document_title: Libation

question_text: the most significant religious act in greek ritual was

text: Libation

text: Religious practice[edit]

text: Historical[edit]

text: Ancient Egypt[edit]

text: Libation was part of ancient Egyptian society where it was a drink offering to honor and please the various divinities, sacred ancestors, humans present and not present, as well as the environment.[citation needed] It is suggested that libation originated somewhere in the upper Nile Valley and spread out to other regions of Africa and the world.[1][2] According to Ayi Kwei Armah, “[t]his legend explains the rise of a propitiatory custom found everywhere on the African continent: libation, the pouring of alcohol or other drinks as offerings to ancestors and divinities.” [3]

text: Ancient Israel[edit]

text: Libations were part of ancient Judaism and are mentioned in the Bible:

text: And Jacob set up a Pillar in the place where he had spoken with him, a Pillar of Stone; and he poured out a drink offering on it, and poured oil on it.

text: Isaiah uses libation as a metaphor when describing the end of the Suffering Servant figure who "poured out his life unto death". (53:12)

text: Ancient Greece[edit]

text: Libation was a central and vital aspect of ancient Greek religion, and one of the simplest and most common forms of religious practice.[4] It is one of the basic religious acts that define piety in ancient Greece, dating back to the Bronze Age and even prehistoric Greece.[5] Libations were a part of daily life, and the pious might perform them every day in the morning and evening, as well as to begin meals.[6] A libation most often consisted of mixed wine and water, but could also be unmixed wine, honey, oil, water, or milk.[7]

text: The form of libation called spondē is typically the ritualized pouring of wine from a jug or bowl held in the hand. The most common ritual was to pour the liquid from an oinochoē (wine jug) into a phiale, a shallow bowl designed for the purpose. After wine was poured from the phiale, the remainder of the contents was drunk by the celebrant.[8] A libation is poured any time wine is to be drunk, a practice that is recorded as early as the Homeric epics. The etiquette of the symposium required that when the first bowl (krater) of wine was served, a libation was made to Zeus and the Olympian gods. Heroes received a libation from the second krater served, and "Zeus the Finisher" (Zeus Teleios) from the third, which was supposed to be the last. An alternative was to offer a libation from the first bowl to the Agathos Daimon and from the third bowl to Hermes. An individual at the symposium could also make an invocation of and libation to a god of his choice.

text: Libation generally accompanied prayer.[9] The Greeks stood when they prayed, either with their arms uplifted, or in the act of libation with the right arm extended to hold the phiale.[10]

text: In conducting animal sacrifice, wine is poured onto the victim as part of its ritual slaughter and preparation, and then afterwards onto the ash and flames.[11] This scene is commonly depicted in Greek art, which also often shows sacrificers or the gods themselves holding the phiale.[12]

text: The Greek verb spendō (σπένδω), "pour a libation", also "conclude a pact", derives from the Indo-European root *spend-, "make an offering, perform a rite, engage oneself by a ritual act". The noun is spondē or spondai, "libation." In the middle voice, the verb means "enter into an agreement", in the sense that the gods are called to guarantee an action.[13] Blood sacrifice was performed to begin a war; spondai marked the conclusion of hostilities, and is often thus used in the sense of "armistice, treaty." The formula "We the polis have made libation" was a declaration of peace or the "Truce of God", which was observed also when the various city-states came together for the Panhellenic Games, the Olympic Games, or the festivals of the Eleusinian Mysteries: this form of libation is "bloodless, gentle, irrevocable, and final".[12]

text: Libations poured onto the earth are meant for the dead and for the chthonic gods. In the Book of the Dead in the Odyssey, Odysseus digs an offering pit around which he pours in order honey, wine and water. For the form of libation called choē (from IE *gheu-, Greek cheuma, χεῦμα, "that which is poured"[14]), a larger vessel is tipped over and emptied onto the ground for the chthonic gods, who may also receive spondai.[15] Heroes, who were divinized mortals, might receive blood libations if they had participated in the bloodshed of war, as for instance Brasidas the Spartan.[16] In rituals of caring for the dead at their tombs, libations would include milk and honey.[17]

text: The Libation Bearers is the English title of the center tragedy from the Orestes Trilogy of Aeschylus, in reference to the offerings Electra brings to the tomb of her dead father Agamemnon.[12] Sophocles gives one of the most detailed descriptions of libation in Greek literature in Oedipus at Colonus, performed as atonement in the grove of the Eumenides:

text: First, water is fetched from a freshly flowing spring; cauldrons which stand in the sanctuary are garlanded with wool and filled with water and honey; turning towards the east, the sacrificer tips the vessels towards the west; the olive branches which he has been holding in his hand he now strews on the ground at the place where the earth has drunk in the libation; and with a silent prayer he departs, not looking back.[18]

text: Hero of Alexandria described a mechanism for automating the process by using altar fires to force oil from the cups of two statues.[citation needed]

text: Ancient Rome[edit]

text: The English word "libation" derives from the Latin libatio, an act of pouring, from the verb libare, "to taste, sip; pour out, make a libation" (Indo-European root *leib-, "pour, make a libation").[19] In ancient Roman religion, the libation was an act of worship in the form of a liquid offering, most often unmixed wine and perfumed oil.[20] The Roman god Liber Pater ("Father Liber"), later identified with the Greek Dionysus or Bacchus, was the divinity of libamina, "libations," and liba, sacrificial cakes drizzled with honey.[21]

text: In Roman art, the libation is shown performed at an altar, mensa (sacrificial meal table), or tripod. It was the simplest form of sacrifice, and could be a sufficient offering by itself.[22] The introductory rite (praefatio) to an animal sacrifice included an incense and wine libation onto a burning altar.[23] Both emperors and divinities are frequently depicted, especially on coins, pouring libations.[24] Scenes of libation commonly signify the quality of pietas, religious duty or reverence.[25]

text: The libation was part of Roman funeral rites, and may have been the only sacrificial offering at humble funerals.[26] Libations were poured in rituals of caring for the dead (see Parentalia and Caristia), and some tombs were equipped with tubes through which the offerings could be directed to the underground dead.[27]

text: Milk was unusual as a libation at Rome, but was regularly offered to a few deities, particularly those of an archaic nature[28] or those for whom it was a natural complement, such as Rumina, a goddess of birth and childrearing who promoted the flow of breast milk, and Cunina, a tutelary of the cradle.[29] It was offered also to Mercurius Sobrius (the "sober" Mercury), whose cult is well attested in Roman Africa and may have been imported to the city of Rome by an African community.[30]

text: Africa[edit]

text: In African cultures, African traditional religions the ritual of pouring libation is an essential ceremonial tradition and a way of giving homage to the ancestors. Ancestors are not only respected in such cultures, but also invited to participate in all public functions (as are also the gods and God). A prayer is offered in the form of libations, calling the ancestors to attend. The ritual is generally performed by an elder. Although water may be used, the drink is typically some traditional wine (e.g. palm wine), and the libation ritual is accompanied by an invitation (and invocation) to the ancestors, gods and God. In the Volta region of Ghana, water with a mixture of corn flour is also used to pour libation.

text: Libation is also commonly recognized as the break within the famous performance of Agbekor, a ritual dance performed in West African cultures. It is also poured during traditional marriage ceremony, when a child is born and funeral ceremony. Traditional Festivals like Asafotu and Homowo of the Ga Adangbe people of Ghana and Togo. Also during installment of Kings, Queens, Chiefs libation is poured.

text: Americas[edit]

text: In the Quechua and Aymara cultures of the South American Andes, it is common to pour a small amount of one's beverage on the ground before drinking as an offering to the Pachamama, or Mother Earth. This especially holds true when drinking Chicha, an alcoholic beverage unique to this part of the world. The libation ritual is commonly called challa and is performed quite often, usually before meals and during celebrations.

text: Asia[edit]

text: Burmese Buddhism[edit]

text: In Burmese Buddhism, the water libation ceremony, called yay zet cha (ရေစက်ချ), which involves the ceremonial pouring of water from a vessel of water into a vase, drop by drop, concludes most Buddhist ceremonies, including donation celebrations, shinbyu, and feasts. This ceremonial libation is done to share the accrued merit with all other living beings in all 31 planes of existence.[31] The ceremony has three primary prayers: the confession of faith, the pouring of water, and the sharing of merits.[32] While the water is poured, a confession of faith, called the hsu taung imaya dhammanu (ဆုတောင်း ဣမာယ ဓမ္မာနု), is recited and led by the monks.[33]

text: Then, the merit is distributed by the donors (called ahmya wei) by thrice saying the following:[32]

text:
(To all those who can hear), we share our merits with all beings
(Kya kya thahmya), ahmya ahmya ahmya yu daw mu gya ba gon law
((ကြားကြားသမျှ) အမျှ အမျှ အမျှ ယူတော်မူကြပါ ကုန်လော)

text: Afterward, in unison, the participants repeat thrice a declaration of affirmation: thadu (သာဓု, sadhu), Pali for "well done", akin to the Christian use of amen. Afterward, the libated water is poured on soil outside, to return the water to Vasudhara. The earth goddess Vasudhara is invoked to witness these meritorious deeds.[33]

text: Prior to colonial rule, the water libation ceremony was also performed during the crowning of Burmese kings, as part of procedures written in the Raza Thewaka Dipani Kyan, an 1849 text that outlines proper conduct of Burmese kings.[34][35]

text: Although the offering of water to Vasudhara may have pre-Buddhist roots, this ceremony is believed to have been started by King Bimbisara, who poured the libation of water, to share his merit with his ancestors who had become pretas.[36][37][38]

text: This ceremony is also practiced at the end of Thai and Laotian Buddhist rituals to transfer merit, where it is called kruat nam (กรวดน้ำ) and yaat nam respectively.[39]

text: Hinduism[edit]

text: In Hinduism the ritual is part of Tarpan and also performed during Pitru Paksha (Fortnight of the ancestors) falling the Bhadrapada month of Hindu calendar, (Sept-Oct).[40] In India and Nepal, Lord Shiva (also Vishnu and other deities) is offered libation with water by devotees at many temples when they go visit the temple, and on special occasions elaborately with water, milk, yogurt, ghee, honey, and sugar.

text: China[edit]

text: In Chinese customs, rice wine or tea is poured in front of an altar or tombstone horizontally from right to left with both hands as an offering to gods and in honour of deceased. The offering is usually placed on the altar for a while before being offered in libation. In more elaborate ceremonies honouring deities, the libation may be done over the burning paper offerings; whereas for the deceased, the wine is only poured onto the ground.

text: Japan[edit]

text: In Shinto, the practice of libation and the drink offered is called Miki (神酒), lit. "The Liquor of the Gods". At a ceremony at a Shinto shrine, it is usually done with sake, but at a household shrine, one may substitute fresh water which can be changed every morning. It is served in a white porcelain or metal cup without any decoration.

text: Siberian Shamanism[edit]

text: Shamanism among Siberian peoples exhibits the great diversity characteristic of shamanism in general.[41] Among several peoples near the Altai Mountains, the new drum of a shaman must go through a special ritual. This is regarded as “enlivening the drum”: the tree and the deer who gave their wood and skin for the new drum narrate their whole lives and promise to the shaman that they will serve him. The ritual itself is a libation: beer is poured onto the skin and wood of the drum, and these materials “come to life” and speak with the voice of the shaman in the name of the tree and the deer. Among the Tubalar, moreover, the shaman imitates the voice of the animal, and its behaviour as well.[42]

text: Modern customs[edit]

text: In Cuba a widespread custom is to spill a drop or two of rum from one's glass while saying "para los santos" (‘for the Saints’) This custom is similar to the practise of Visayans living on Mindanao, the Philippines, where they spill a capful of rum as soon as the bottle is opened while saying "para sa yawa" ('for the Devil').[43]

text: In Russia and surrounding countries, it is an old tradition to pour vodka onto a grave, an act possibly connected with dziady custom.

text: In the contemporary United States, there is a tradition of pouring libations of malt liquor from a forty before drinking, which is particularly associated with African-American rappers. This is referred to as "tipping" to one's [dead] homies (friends),[44] or "pouring one out".[45] This is referenced in various songs, such as the 1993 "Gangsta Lean (This Is For My Homies)" by DRS ("I tip my 40 to your memory"), and sometimes accompanied by ritual expressions such as "One for me, and one for my homies" as well as the 1994 song "Pour Out a Little Liquor" by 2Pac. This is occasionally parodied, as in Austin Powers: The Spy Who Shagged Me (1999).[46]


document_title: Montgomery bus boycott

question_text: when did the supreme court rule that bus segregation was unconstitutional

root

text: Montgomery bus boycott

text: Events leading up to the bus boycott[edit]

text: Prior to the bus boycott, Jim Crow laws enabled the racial segregation of the Montgomery Bus Line. As a result of this segregation African Americans were not hired as drivers, were forced to ride in the back of the bus, and were frequently ordered to surrender their seats to white people even though black passengers made up 75% of the bus system's riders.[2]

tag: p

text: African-American passengers were also attacked by bus drivers and shortchanged and left stranded after paying their fares.[3] A number of reasons have been given for why bus drivers acted in this manner, including racism,[4] frustrations over labor disputes and labor conditions, and increased animosity towards blacks in reaction to the 1954 Brown decision, with many of the drivers joining the White Citizens Councils as a result of the decision.[5]

text: The boycott also took place within a larger statewide and national movement for civil rights, including court cases such as Morgan v. Virginia, the earlier Baton Rouge bus boycott, and the arrest of Claudette Colvin.

text: Morgan v. Virginia decision[edit]

text: The NAACP had accepted and litigated other cases, including that of Irene Morgan in 1946, which resulted in a victory in the U.S. Supreme Court on the grounds that segregated interstate bus lines violated the Commerce Clause.[6] That victory, however, overturned state segregation laws only insofar as they applied to travel in interstate commerce, such as interstate bus travel,[7] and Southern bus companies immediately circumvented the Morgan ruling by instituting their own Jim Crow regulations.[8] Further incidents continued to take place in Montgomery, including the arrest for disorderly conduct in May 1951 of Lillie Mae Bradford, who refused to leave the white passengers' section until the bus driver corrected an incorrect charge on her transfer ticket.[9]

text: Baton Rouge bus boycott[edit]

text: On February 25, 1953, the Baton Rouge, Louisiana city-parish council passed Ordinance 222, after the city saw protesting from African-Americans when the council raised the city's bus fares.[10] The ordinance abolished race-based reserved seating requirements and allowed the admission of African-Americans in the front sections of city buses if there were no white passengers present, but still required African-Americans to enter from the rear, rather than the front of the buses.[11] However, the ordinance was largely unenforced by the city bus drivers. The drivers later went on strike after city authorities refused to arrest Rev. T.J. Jemison for sitting in a front row.[12] Four days after the strike began, Louisiana Attorney General and former Baton Rouge mayor Fred S. LeBlanc declared the ordinance unconstitutional under Louisiana state law.[11] This led Rev. Jemison to organize what historians believe to be the first bus boycott of the civil rights movement.[13] The boycott ended after eight days when an agreement was reached to only retain the first two front and back rows as racially reserved seating areas.[10]

text: Arrest of Claudette Colvin[edit]

text: Black activists had begun to build a case to challenge state bus segregation laws around the arrest of a 15-year-old girl, Claudette Colvin, a student at Booker T. Washington High School in Montgomery. On March 2, 1955, Colvin was handcuffed, arrested and forcibly removed from a public bus when she refused to give up her seat to a white man. At the time, Colvin was an active member in the NAACP Youth Council; Rosa Parks was an advisor.[14] Colvin's legal case formed the core of Browder v. Gayle, which ended the Montgomery bus boycott when the Supreme Court ruled on it in December 1956.

text: Keys v. Carolina Coach Co. decision[edit]

text: In November 1955, just three weeks before Parks' defiance of Jim Crow laws in Montgomery, the Interstate Commerce Commission, in response to a complaint filed by Women's Army Corps private Sarah Keys, closed the legal loophole left by the Morgan ruling in a landmark case known as Keys v. Carolina Coach Co..[15] The ICC prohibited individual carriers from imposing their own segregation rules on interstate travelers, declaring that to do so was a violation of the anti-discrimination provision of the Interstate Commerce Act. But neither the Supreme Court's Morgan ruling nor the ICC's Keys ruling addressed the matter of Jim Crow travel within the individual states.[citation needed]

text: History[edit]

text: Under the system of segregation used on Montgomery buses, the ten front seats were reserved for whites at all times. The ten back seats were supposed to be reserved for blacks at all times. The middle section of the bus consisted of sixteen unreserved seats for whites and blacks on a segregated basis.[16] Whites filled the middle seats from the front to back, and blacks filled seats from the back to front until the bus was full. If other black people boarded the bus, they were required to stand. If another white person boarded the bus, then everyone in the black row nearest the front had to get up and stand, so that a new row for white people could be created; it was illegal for whites and blacks to sit next to each other. When Rosa Parks refused to give up her seat for a white person, she was sitting in the first row of the middle section.[17]

text: Often when boarding the buses, black people were required to pay at the front, get off, and reenter the bus through a separate door at the back.[18] Occasionally, bus drivers would drive away before black passengers were able to reboard.[19] National City Lines owned the Montgomery Bus Line at the time of the Montgomery bus boycott.[20]

text: Rosa Parks[edit]

text: Rosa Parks (February 4, 1913 – October 24, 2005) was a seamstress by profession; she was also the secretary for the Montgomery chapter of the NAACP. Twelve years before her history-making arrest, Parks was stopped from boarding a city bus by driver James F. Blake, who ordered her to board at the back door and then drove off without her. Parks vowed never again to ride a bus driven by Blake. As a member of the NAACP, Parks was an investigator assigned to cases of sexual assault. In 1945, she was sent to Abbeville, Alabama, to investigate the gang rape of Recy Taylor. The protest that arose around the Taylor case was the first instance of a nationwide civil rights protest, and it laid the groundwork for the Montgomery bus boycott.[21]

text: In 1955, Parks completed a course in "Race Relations" at the Highlander Folk School in Tennessee, where nonviolent civil disobedience had been discussed as a tactic. On December 1, 1955, Parks was sitting in the foremost row in which black people could sit (in the middle section). When a white man boarded the bus, the bus driver told everyone in her row to move back. At that moment, Parks realized that she was again on a bus driven by Blake. While all of the other black people in her row complied, Parks refused, and she was arrested[22] for failing to obey the driver's seat assignments, as city ordinances did not explicitly mandate segregation but did give the bus driver authority to assign seats. Found guilty on December 5,[23] Parks was fined $10 plus a court cost of $4[24], and she appealed.[citation needed]

text: E. D. Nixon[edit]

text: This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2012) (Learn how and when to remove this template message)

tag: table

text: Some action against segregation had been in the works for some time before Parks' arrest, under the leadership of E. D. Nixon, president of the local NAACP chapter and a member of the Brotherhood of Sleeping Car Porters. Nixon intended that her arrest be a test case to allow Montgomery's black citizens to challenge segregation on the city's public buses. With this goal, community leaders had been waiting for the right person to be arrested, a person who would anger the black community into action, who would agree to test the segregation laws in court, and who, most importantly, was "above reproach". When Colvin was arrested in March 1955, Nixon thought he had found the perfect person, but the teenager turned out to be pregnant. Nixon later explained, "I had to be sure that I had somebody I could win with." Parks was a good candidate because of her employment and marital status, along with her good standing in the community.

text: Between Parks' arrest and trial, Nixon organized a meeting of local ministers at Martin Luther King Jr.'s church. Though Nixon could not attend the meeting because of his work schedule, he arranged that no election of a leader for the proposed boycott would take place until his return. When he returned, he caucused with Ralph Abernathy and Rev. E.N. French to name the association to lead the boycott to the city (they selected the "Montgomery Improvement Association", "MIA"), and they selected King (Nixon's choice) to lead the boycott. Nixon wanted King to lead the boycott because the young minister was new to Montgomery and the city fathers had not had time to intimidate him. At a subsequent, larger meeting of ministers, Nixon's agenda was threatened by the clergymen's reluctance to support the campaign. Nixon was indignant, pointing out that their poor congregations worked to put money into the collection plates so these ministers could live well, and when those congregations needed the clergy to stand up for them, those comfortable ministers refused to do so. Nixon threatened to reveal the ministers' cowardice to the black community, and King spoke up, denying he was afraid to support the boycott. King agreed to lead the MIA, and Nixon was elected its treasurer.

text: Boycott[edit]

text: On the night of Rosa Parks' arrest, the Women's Political Council, led by Jo Ann Robinson, printed and circulated a flyer[25] throughout Montgomery's black community that read as follows:

text: Another woman has been arrested and thrown in jail because she refused to get up out of her seat on the bus for a white person to sit down. It is the second time since the Claudette Colvin case that a Negro woman has been arrested for the same thing. This has to be stopped. Negroes have rights too, for if Negroes did not ride the buses, they could not operate. Three-fourths of the riders are Negro, yet we are arrested, or have to stand over empty seats. If we do not do something to stop these arrests, they will continue. The next time it may be you, or your daughter, or mother. This woman's case will come up on Monday. We are, therefore, asking every Negro to stay off the buses Monday in protest of the arrest and trial. Don't ride the buses to work, to town, to school, or anywhere on Monday. You can afford to stay out of school for one day if you have no other way to go except by bus. You can also afford to stay out of town for one day. If you work, take a cab, or walk. But please, children and grown-ups, don't ride the bus at all on Monday. Please stay off all buses Monday.[24]

text: The next morning there was a meeting led by the new MIA head, King, where a group of 16 to 18 people gathered at the Mt. Zion Church to discuss boycott strategies. At that time Rosa Parks was introduced but not asked to speak, despite a standing ovation and calls from the crowd for her to speak; when she asked if she should say something, the reply was, "Why, you've said enough."[26] A citywide boycott of public transit was proposed to demand a fixed dividing line for the segregated sections of the buses. Such a line would have meant that if the white section of the bus was oversubscribed, whites would have to stand; blacks would not be forced to give up their seats to whites.[citation needed]

text: This demand was a compromise for the leaders of the boycott, who believed that the city of Montgomery would be more likely to accept it rather than a demand for a full integration of the buses. In this respect, the MIA leaders followed the pattern of 1950s boycott campaigns in the Deep South, including the successful boycott a few years earlier of service stations in Mississippi for refusing to provide restrooms for blacks. The organizer of that campaign, T. R. M. Howard of the Regional Council of Negro Leadership, had spoken on the brutal slaying of Emmett Till as King's guest at the Dexter Avenue Baptist Church only four days before Parks's arrest. Parks was in the audience and later said that Emmett Till was on her mind when she refused to give up her seat.[27]

text: The MIA's demand for a fixed dividing line was to be supplemented by a requirement that all bus passengers receive courteous treatment by bus operators, be seated on a first-come, first-served basis, and that blacks be employed as bus drivers.[28] The proposal was passed, and the boycott was to commence the following Monday. To publicize the impending boycott it was advertised at black churches throughout Montgomery the following Sunday.[citation needed]

text: On Saturday, December 3, it was evident that the black community would support the boycott, and very few blacks rode the buses that day. On December 5, a mass meeting was held at the Holt Street Baptist Church to determine if the protest would continue.[29] Given twenty minutes notice, King gave a speech[30] asking for a bus boycott and attendees enthusiastically agreed. Starting December 7, Hoover's FBI noted the "agitation among negroes" and tried to find "derogatory information" about King.[31]

text: The boycott proved extremely effective, with enough riders lost to the city transit system to cause serious economic distress. Martin Luther King later wrote "[a] miracle had taken place." Instead of riding buses, boycotters organized a system of carpools, with car owners volunteering their vehicles or themselves driving people to various destinations. Some white housewives also drove their black domestic servants to work. When the city pressured local insurance companies to stop insuring cars used in the carpools, the boycott leaders arranged policies at Lloyd's of London.[citation needed]

text: Black taxi drivers charged ten cents per ride, a fare equal to the cost to ride the bus, in support of the boycott. When word of this reached city officials on December 8, the order went out to fine any cab driver who charged a rider less than 45 cents. In addition to using private motor vehicles, some people used non-motorized means to get around, such as cycling, walking, or even riding mules or driving horse-drawn buggies. Some people also hitchhiked. During rush hours, sidewalks were often crowded. As the buses received few, if any, passengers, their officials asked the City Commission to allow stopping service to black communities.[32] Across the nation, black churches raised money to support the boycott and collected new and slightly used shoes to replace the tattered footwear of Montgomery's black citizens, many of whom walked everywhere rather than ride the buses and submit to Jim Crow laws.[citation needed]

text: In response, opposing whites swelled the ranks of the White Citizens' Council, the membership of which doubled during the course of the boycott. The councils sometimes resorted to violence: King's and Abernathy's houses were firebombed, as were four black Baptist churches. Boycotters were often physically attacked. After the attack at King's house, he gave a speech to the 300 angry African Americans who had gathered outside. He said:

text: If you have weapons, take them home; if you do not have them, please do not seek to get them. We cannot solve this problem through retaliatory violence. We must meet violence with nonviolence. Remember the words of Jesus: "He who lives by the sword will perish by the sword". We must love our white brothers, no matter what they do to us. We must make them know that we love them. Jesus still cries out in words that echo across the centuries: "Love your enemies; bless them that curse you; pray for them that despitefully use you". This is what we must live by. We must meet hate with love. Remember, if I am stopped, this movement will not stop, because God is with the movement. Go home with this glowing faith and this radiant assurance.[33]

text: King and 89 other boycott leaders and carpool drivers were indicted[34] for conspiring to interfere with a business under a 1921 ordinance.[35] Rather than wait to be arrested, they turned themselves in as an act of defiance.[citation needed]

text: King was ordered to pay a $500 fine or serve 386 days in jail. He ended up spending two weeks in jail. The move backfired by bringing national attention to the protest. King commented on the arrest by saying: "I was proud of my crime. It was the crime of joining my people in a nonviolent protest against injustice."[36]

text: Also important during the bus boycott were grass-roots activist groups that helped to catalyze both fund-raising and morale. Groups such as the Club from Nowhere helped to sustain the boycott by finding new ways of raising money and offering support to boycott participants.[37] Many members of these organizations were women and their contributions to the effort have been described by some as essential to the success of the bus boycott.[38][39]

text: Victory[edit]

text: Pressure increased across the country. The related civil suit was heard in federal district court and, on June 4, 1956, the court ruled in Browder v. Gayle (1956) that Alabama's racial segregation laws for buses were unconstitutional. As the state appealed the decision, the boycott continued. The case moved on to the United States Supreme Court. On November 13, 1956, the Supreme Court upheld the district court's ruling, ruling that segregation on public buses and transportation was against the law.

text: The boycott officially ended December 20, 1956, after 381 days. The city passed an ordinance authorizing black bus passengers to sit virtually anywhere they chose on buses. The Montgomery bus boycott resounded far beyond the desegregation of public buses. It stimulated activism and participation from the South in the national Civil Rights Movement and gave King national attention as a rising leader.[41][42]

text: Aftermath[edit]

text: White backlash against the court victory was quick, brutal, and, in the short-term, effective.[43][44] Two days after the inauguration of desegregated seating, someone fired a shotgun through the front door of Martin Luther King's home. A day later, on Christmas Eve, white men attacked a black teenager as she exited a bus. Four days after that, two buses were fired upon by snipers. In one sniper incident, a pregnant woman was shot in both legs. On January 10, 1957, bombs destroyed five black churches and the home of Reverend Robert S. Graetz, one of the few white Montgomerians who had publicly sided with the MIA.[45][46]

text: The City suspended bus service for several weeks on account of the violence. According to legal historian Randall Kennedy, "When the violence subsided and service was restored, many black Montgomerians enjoyed their newly recognized right only abstractly... In practically every other setting, Montgomery remained overwhelmingly segregated…"[46] On January 23, a group of Klansmen (who would later be charged for the bombings) lynched a black man, Willie Edwards, on the pretext that he was dating a white woman.[47]

text: The City's elite moved to strengthen segregation in other areas, and in March 1957 passed an ordinance making it "unlawful for white and colored persons to play together, or, in company with each other . . . in any game of cards, dice, dominoes, checkers, pool, billiards, softball, basketball, baseball, football, golf, track, and at swimming pools, beaches, lakes or ponds or any other game or games or athletic contests, either indoors or outdoors."[46]

text: Later in the year, Montgomery police charged seven Klansmen with the bombings, but all of the defendants were acquitted. About the same time, the Alabama Supreme Court ruled against Martin Luther King's appeal of his "illegal boycott" conviction.[48] Rosa Parks left Montgomery due to death threats and employment blacklisting.[49] According to Charles Silberman, "by 1963, most Negroes in Montgomery had returned to the old custom of riding in the back of the bus."[50]

text: Participants[edit]

text: People[edit]

text:
Ralph Abernathy
Hugo Black
James F. Blake
Aurelia Browder
Mary Fair Burks
Johnnie Carr
Claudette Colvin
Clifford Durr
Mildred Fahrni[51]
Georgia Gilmore
Robert Graetz
Fred Gray
Grover C. Hall, Jr.
Coretta Scott King
Martin Luther King Jr.
Theodora Lacey
Edgar Nixon
Rosa Parks
Mother Pollard
Jo Ann Robinson
Bayard Rustin
Nate Singleton
Glenn Smiley
Mary Louise Smith

tag: ul

text: Organizations[edit]

text:
Committee for Nonviolent Integration
Fellowship of Reconciliation
Georgia Gilmore
Men of Montgomery
Montgomery Improvement Association
National Association for the Advancement of Colored People
Women's Political Council


document_title: Standing committee (United States Congress)

question_text: what are the standing committees of the house and senate

text: This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (February 2015) (Learn how and when to remove this template message)

text: In the United States Congress, standing committees are permanent legislative panels established by the United States House of Representatives and United States Senate rules. (House Rule X, Senate Rule XXV.) Because they have legislative jurisdiction, standing committees consider bills and issues and recommend measures for consideration by their respective chambers. They also have oversight responsibility to monitor agencies, programs, and activities within their jurisdictions, and in some cases in areas that cut across committee jurisdictions. Due to their permanent nature, these committees exist beyond the adjournment of each two-year meeting of Congress.

text: Most standing committees recommend funding levels—authorizations—for government operations and for new and existing programs. A few have other functions. For example, the Appropriations Committees recommend legislation to provide budget authority for federal agencies and programs. The Budget Committees establish aggregate levels for total spending and revenue that serve as guidelines for the work of the authorizing and appropriating panels. Committees also provide oversight of federal agencies and programs.

text: The Legislative Reorganization Act of 1946 greatly reduced the number of committees. The membership of each committee is adopted at the beginning of each Congress, usually by adoption of a formal resolution. Each committee is assigned its own staff to assist with its legislative, investigative, and research functions. Several committees divide their work into sub units called subcommittees.

text: Committee sizes range from 6 to 50 members per committee. In the House, one person may not serve on more than two standing committees and four subcommittees at one time, though waivers can be granted to serve on additional committees. Also in the House, the House Republican Steering Committee assigns Republican representatives to their committee(s),[1][2] while the Steering and Policy Committee is in charge of assigning Democratic representatives to committees.[3] The Senate follows similar procedures, with senators being limited to no more than three full committees and five sub-committees.

text: Current number of standing committees[edit]

text: As of June 17, 2017, the Senate had[update] 16 standing committees and the House had 20 standing committees.[4] (The count is for standing committees only and does not include select or special committees or joint committees. See those articles for that information.)


document_title: Timeline of knowledge about galaxies, clusters of galaxies, and large-scale structure

question_text: when were galaxies discovered to be large separate structures in the universe

text: Pre-20th century[edit]

text:
5th century BC — Democritus proposes that the bright band in the night sky known as the Milky Way might consist of stars,
4th century BC — Aristotle believes the Milky Way to be caused by "the ignition of the fiery exhalation of some stars which were large, numerous and close together" and that the "ignition takes place in the upper part of the atmosphere, in the region of the world which is continuous with the heavenly motions",[1]
964 — Abd al-Rahman al-Sufi (Azophi), a Persian astronomer, makes the first recorded observations of the Andromeda Galaxy[2] and the Large Magellanic Cloud[3][4] in his Book of Fixed Stars, and which are the first galaxies other than the Milky Way to be observed from Earth,
11th century — Abū Rayḥān al-Bīrūnī, another Persian astronomer, describes the Milky Way galaxy as a collection of numerous nebulous stars,[5]
11th century — Alhazen (Ibn al-Haytham), an Arabian astronomer, refutes Aristotle's theory on the Milky Way by making the first attempt at observing and measuring the Milky Way's parallax,[6] and he thus "determined that because the Milky Way had no parallax, it was very remote from the Earth and did not belong to the atmosphere",[7]
12th century — Avempace (Ibn Bajjah) of Islamic Spain proposes the Milky Way to be made up of many stars but that it appears to be a continuous image due to the effect of refraction in the Earth's atmosphere,[1]
14th century — Ibn Qayyim Al-Jawziyya of Syria proposes the Milky Way galaxy to be "a myriad of tiny stars packed together in the sphere of the fixed stars" and that these stars are larger than planets,[8]
1521 — Ferdinand Magellan observes the Magellanic Clouds during his circumnavigating expedition,
1610 — Galileo Galilei uses a telescope to determine that the bright band on the sky, the "Milky Way", is composed of many faint stars
1750 — Thomas Wright discusses galaxies and the shape of the Milky Way,
1755 — Drawing on Wright's work, Immanuel Kant conjectures that the galaxy is a rotating disk of stars held together by gravity, and that the nebulae are separate such galaxies; he calls them Island Universes
1785 — William Herschel carried the first attempt to describe the shape of the Milky Way and the position of the Sun in it by carefully counting the number of stars in different regions of the sky. He produced a diagram of the shape of the galaxy with the solar system close to the center.
1845 — Lord Rosse discovers a nebula with a distinct spiral shape

text: Early 20th century[edit]

text:
1918 — Harlow Shapley demonstrates that globular clusters are arranged in a spheroid or halo whose center is not the Earth, and decides, correctly, that its center is the Galactic Center of the galaxy,
26 April 1920 — Harlow Shapley and Heber Curtis debate whether or not the Andromeda Nebula is within the Milky Way,
1923 — Edwin Hubble resolves the Shapley–Curtis debate by finding Cepheids in the Andromeda Galaxy
1930 — Robert Trumpler uses open cluster observations to quantify the absorption of light by interstellar dust in the galactic plane; this absorption had plagued earlier models of the Milky Way,
1932 — Karl Guthe Jansky discovers radio noise from the center of the Milky Way,
1933 — Fritz Zwicky applies the virial theorem to the Coma Cluster and obtains evidence for unseen mass,
1936 — Edwin Hubble introduces the spiral, barred spiral, elliptical, and irregular galaxy classifications,
1939 — Grote Reber discovers the radio source Cygnus A,
1943 — Carl Keenan Seyfert identifies six spiral galaxies with unusually broad emission lines, named Seyfert galaxies,
1949 — J. G. Bolton, G. J. Stanley, and O. B. Slee identify NGC 4486 (M87) and NGC 5128 as extragalactic radio sources,

text: Mid-20th century[edit]

text:
1953 — Gérard de Vaucouleurs discovers that the galaxies within approximately 200 million light-years of the Virgo Cluster are confined to a giant supercluster disk,
1954 — Walter Baade and Rudolph Minkowski identify the extragalactic optical counterpart of the radio source Cygnus A,
1959 — Hundreds of radio sources are detected by the Cambridge Interferometer which produces the 3C catalogue. Many of these are later found to be distant quasars and radio galaxies
1960 — Thomas Matthews determines the radio position of the 3C source 3C 48 to within 5",
1960 — Allan Sandage optically studies 3C 48 and observes an unusual blue quasistellar object,
1962 — Cyril Hazard, M. B. Mackey, and A. J. Shimmins use lunar occultations to determine a precise position for the quasar 3C 273 and deduce that it is a double source,
1962 — Olin Eggen, Donald Lynden-Bell, and Allan Sandage theorize galaxy formation by a single (relatively) rapid monolithic collapse, with the halo forming first, followed by the disk.
1963 — Maarten Schmidt identifies the redshifted Balmer lines from the quasar 3C 273
1973 — Jeremiah Ostriker and James Peebles discover that the amount of visible matter in the disks of typical spiral galaxies is not enough for Newtonian gravitation to keep the disks from flying apart or drastically changing shape,
1973 — Donald Gudehus finds that the diameters of the brightest cluster galaxies have increased due to merging, the diameters of the faintest cluster galaxies have decreased due to tidal distention, and that the Virgo cluster has a substantial peculiar velocity,
1974 — B. L. Fanaroff and J. M. Riley distinguish between edge-darkened (FR I) and edge-brightened (FR II) radio sources,
1976 — Sandra Faber and Robert Jackson discover the Faber-Jackson relation between the luminosity of an elliptical galaxy and the velocity dispersion in its center. In 1991 the relation is revised by Donald Gudehus,
1977 — R. Brent Tully and Richard Fisher publish the Tully–Fisher relation between the luminosity of an isolated spiral galaxy and the velocity of the flat part of its rotation curve,
1978 — Steve Gregory and Laird Thompson describe the Coma supercluster,
1978 — Donald Gudehus finds evidence that clusters of galaxies are moving at several hundred kilometers per second relative to the cosmic microwave background radiation,
1978 — Vera Rubin, Kent Ford, N. Thonnard, and Albert Bosma measure the rotation curves of several spiral galaxies and find significant deviations from what is predicted by the Newtonian gravitation of visible stars,
1978 — Leonard Searle and Robert Zinn theorize that galaxy formation occurs through the merger of smaller groups.

text: Late 20th century[edit]

text: This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (April 2009) (Learn how and when to remove this template message)

text:
1981 — Robert Kirshner, August Oemler, Paul Schechter, and Stephen Shectman find evidence for a giant void in Boötes with a diameter of approximately 100 million light years,
1985 — Robert Antonucci and J. Miller discover that the Seyfert II galaxy NGC 1068 has broad lines which can only be seen in polarized reflected light,
1986 — Amos Yahil, David Walker, and Michael Rowan-Robinson find that the direction of the IRAS galaxy density dipole agrees with the direction of the cosmic microwave background temperature dipole,
1987 — David Burstein, Roger Davies, Alan Dressler, Sandra Faber, Donald Lynden-Bell, R. J. Terlevich, and Gary Wegner claim that a large group of galaxies within about 200 million light years of the Milky Way are moving together towards the "Great Attractor" in the direction of Hydra and Centaurus,
1987 — R. Brent Tully discovers the Pisces–Cetus Supercluster Complex, a structure one billion light years long and 150 million light years wide,
1989 — Margaret Geller and John Huchra discover the "Great Wall", a sheet of galaxies more than 500 million light years long and 200 million wide, but only 15 million light years thick,
1990 — Michael Rowan-Robinson and Tom Broadhurst discover that the IRAS galaxy IRAS F10214+4724 is the brightest known object in the Universe,
1991 — Donald Gudehus discovers a serious systematic bias in certain cluster galaxy data (surface brightness vs. radius parameter, and the

D

n

{\displaystyle D_{n}}

 method) which affect galaxy distances and evolutionary history; he devises a new distance indicator, the reduced galaxian radius parameter,

r

g

{\displaystyle r_{g}}

, which is free of biases,
1992 — First detection of large-scale structure in the Cosmic microwave background indicating the seeds of the first clusters of galaxies in the early Universe
1995 — First detection of small-scale structure in the cosmic microwave background
1995 — Hubble Deep Field survey of galaxies in field 144 arc seconds across.
1998 — The 2dF Galaxy Redshift Survey maps the large-scale structure in a section of the Universe close to the Milky Way
1998 — Hubble Deep Field South
1998 — Discovery of accelerating universe
2000 — Data from several cosmic microwave background experiments give strong evidence that the Universe is "flat" (space is not curved, although space-time is), with important implications for the formation of large-scale structure

text: Early 21st century[edit]

text:
2001 — First data release from the ongoing Sloan Digital Sky Survey
2004 — The European Southern Observatory discovers Abell 1835 IR1916, the most distant galaxy yet seen from Earth.
2004 — The Arcminute Microkelvin Imager begins to map the distribution of distant clusters of galaxies
2005 — Spitzer Space Telescope data confirm what had been considered likely since the early 1990s from radio telescope data, i.e., that the Milky Way Galaxy is a barred spiral galaxy.[9][10][11]
2012 — Astronomers report the discovery of the most distant dwarf galaxy yet found, approximately 10 billion light-years away.[12]
2012 — The Huge-LQG, a large quasar group, one of the largest known structures in the universe, is discovered.[13]
2013 — The galaxy Z8 GND 5296 is confirmed by spectroscopy to be one of the most distant galaxies found up to this time. Formed just 700 million years after the Big Bang, expansion of the universe has carried it to its current location, about 30 billion light years away from Earth.
2013 — The Hercules–Corona Borealis Great Wall, a massive galaxy filament and the largest known structure in the universe, was discovered through gamma-ray burst mapping.[14][15][16]


document_title: Callie Torres

question_text: who is callie dating on grey's anatomy

question_tokens: who is callie dating on grey 's anatomy

text: Callie Torres

text: Storylines[edit]

text: Callie Torres is introduced as an orthopedic surgery resident with a crush on intern George O'Malley (T.R. Knight). The two start a relationship and Torres moves into Meredith's house (Meredith and Izzie Stevens lived there as well). Uneased at the way their relationship is progressing, O'Malley confronts her, which results in her moving into a hotel. In the aftermath of the breakup, she meets and has a one-night stand with plastic surgeon Mark Sloan (Eric Dane), who becomes her friend. Torres and O'Malley subsequently reconcile, and the two marry in Las Vegas on the spur of the moment. Confused, O'Malley confides to fellow intern Izzie Stevens (Katherine Heigl), about his relationship woes over alcohol, leading to a drunken sexual encounter between the two. Torres remains unaware of his sexual encounter, but becomes increasingly suspicious, and announces her desire to conceive a child. Shortly thereafter, however, she discovers that O'Malley has been unfaithful, ending their marriage. Though her personal life is troubled, Torres thrives professionally when she is appointed as the hospital's Chief Resident. However, she soon begins struggling to cope in the role, and is demoted soon thereafter.

text: Torres forms a friendship with Erica Hahn (Brooke Smith), the hospital's new chief of cardiothoracic surgery. The two embark on a relationship when Hahn jokingly kisses Torres in an elevator to tease Sloan. Neither has been with another woman before, and Torres struggles with her bisexuality, and cheats on her with Sloan. She is initially forgiven, but after an argument about work, Hahn resigns from Seattle Grace, ending their relationship. New pediatric surgeon Arizona Robbins (Jessica Capshaw) kisses her, and they begin dating. Their relationship is tested when Torres comes out to her father, Carlos, who disowns her, cutting her off both financially and emotionally. When she is not granted a position as an attending, she publicly berates the chief, Richard Webber (James Pickens, Jr.), and resigns from her post. She begins working at the neighboring Mercy West hospital, and when the two hospitals merge, she is promoted to an attending surgeon. Torres' father returns yet again, to condemn her, but ultimately accepts her sexuality. When Torres explains her desire to bear children, Robbins expresses disappointment, and their differing stances on the matter lead them to break up. Soon thereafter, a gunman commits mass murder at the hospital, and, relieved to have survived, Torres and Robbins reconcile, with Robbins agreeing that they should have children together.

text: When Robbins wins a prestigious grant to help treat children in Malawi, a disgruntled Torres agrees to move there with her, as the position will last for three years; however, her lack of enthusiasm causes Robbins to end their relationship and go without her. A heartbroken Torres moves in with Sloan, and they have a one-night stand. Robbins returns for Torres, apologizing and hoping for forgiveness, but Torres initially rebuffs her, and later reveals that she is pregnant with Sloan's baby. Robbins agrees to raise the child with her, but dislikes that Sloan will be a permanent part of their lives. En route a weekend getaway, Robbins proposes to Torres, but before Torres can reply, the two are in a head-on collision with a truck, leaving Torres severely injured. In an attempt to save her life, obstetrician-gynecologist Addison Montgomery (Kate Walsh), delivers Torres's premature baby. Torres survives, and agrees to marry Robbins. After twelve weeks of recovery, both Torres and baby Sofia are discharged and allowed to leave the hospital. Torres recovers from her surgery, but deals with the backlash from her mother (Gina Gallego)'s attitude toward the wedding and her granddaughter. Torres and Robbins marry in a garden after the minister's wife falls ill, with general surgeon Miranda Bailey (Chandra Wilson), officiating the ceremony.

text: In the aftermath of a plane accident that killed Sloan and Lexie Grey (Chyler Leigh), Torres is forced to make the decision to have Alex Karev amputate Robbins's left leg to keep her alive, which puts strains on their relationship. The hospital is sued and eventually found guilty of negligence. Each victim, including Robbins, Derek Shepherd (Patrick Dempsey), Meredith Grey (Ellen Pompeo), and Cristina Yang (Sandra Oh) must receive $15 million of compensation, which leads the hospital to a near bankruptcy as the insurances refuse to pay. These doctors and Torres buy the hospital with the help of the Harper-Avery Foundation to prevent it from closing, and each become members of the new directing board. Everything seems to go well for Arizona and Callie, until a big storm hits the hospital now named after Mark Sloan and Lexie Grey. In the last episode of the ninth season, Arizona cheats on Callie with Dr. Lauren Boswell (Hilarie Burton). Callie soon discovers Arizona's infidelity by seeing her wedding ring pinned to Dr. Boswell's scrub top. In the aftermath of discovery, both Callie and Arizona share their feelings and it is revealed that Arizona does not forgive Callie for making the call to amputate her leg. Callie is heartbroken and Arizona states that Callie lost nothing in the crash, to which Callie responds that she, apparently, lost Arizona.

text: Callie takes Sofia to live with Meredith and Derek, Zola, and Baby Bailey for a while before kicking Arizona out of their apartment and celebrates by dancing in her underwear. Callie also left Arizona at therapy, telling her that she was the only one that needed to go. After being hit by a lawsuit, her father, Carlos, visits and tells Callie that he cheated on her mother but she took him back and Callie wouldn't be here if her mother didn't give Carlos a second chance. Callie goes to Arizona's apartment and invites her to come back home. Callie finds out during surgery that Arizona was sleeping with Leah, a second year resident, while separated. Although angry at Arizona, Callie still agrees to work things out.

text: After April's failed wedding to Matthew, Callie and Arizona bought a house together, a fresh start, away from the memories of their apartment. They later decided to try again for a second child. However, a trip to the OB/GYN led to the discovery that Callie had developed adhesions in her uterus in the years since Sofia's birth, meaning she could not carry any more babies. After talking about it, the couple agreed to postpone their dream to have another baby until they're on more solid footing as a couple.

text: At the beginning of the eleventh season, Callie and Arizona gave therapy another chance in order to help mend their marriage. They were made to go 30 days living separate lives in the same house with no sexual activity with others or each other. They come close to breaking that rule and had to start over and before the 30 days was up they eventually gave in and connected sexually again. At the end of the 30 days at the final therapy session Arizona realized she couldn't live without Callie; however Callie didn't feel the same and stated she felt suffocated and eventually walked out of therapy, ending their marriage.

text: Callie then goes on to date Penny, who was the resident in charge of Derek Shepard's case. She comes to a dinner party hosted by Meredith, who recognizes her. It is then discovered that Penny is going to be working at Grey Sloan Memorial. When everyone, including Callie, finds out who Penny actually is, they protest and try to get her off of the Grey Sloan Memorial residency line. She manages to stay on the line despite all the other's pleads and wishes.

text: After Callie discloses her desire to move to New York with Penny and take her and Arizona's daughter, Sofia, with her, Arizona takes Callie to court for sole parental custody. Callie loses custody of her daughter to Arizona and Penny moves to New York. After Arizona decides that both Sofia's moms deserve to be happy, Arizona presents Callie with plane tickets and offers a custody agreement that keeps Sofia in Seattle for the summer and then sharing her every other school year and Christmas.

text: Development[edit]

text: Casting and creation[edit]

text:
"As a Latin person, I was really proud to see the ethnic people on the show. I thought, Wow, there are no color limits. Nobody's making comments about how there are African-Americans on the show and an Asian on the show. However, I did think, Where's the Latin person?"

Ramirez on Grey's Anatomy[2]

tag: div

text: Ramirez was seen by ABC executives, in her Broadway performance of Spamalot, which garnered their attention. Due to their admiration, the network offered Ramirez a role on any ABC television series, of her choice, and she chose Grey's Anatomy.[3] Ramirez further explained that at her initial audition, the producers liked her, and had intentions to add her to the show, but did not know who to cast her as.[2] She also said she was in awe of how the executives said, "Pick a show, any show", explaining that it is rare.[4] Shonda Rhimes said: "I was looking for a girlfriend for George, but it was in the infancy stages, so I had no idea what I was looking for." Rhimes built the character around Ramirez after she met her.[5] Ramirez's character was initially given recurring status in the second season and received star billing in the third season, alongside fellow cast member Dane, who portrayed Mark Sloan.[6] Ramirez discusses maintaining relationships with co-stars:

text:
“
The weird thing about working in television is that you only see the people that you're in scenes with. It's not like you're all running around the set together. So if you're going to hang out together, you kind of have to make an effort. And I think people have families, people have lives.[2]
”

text: Characterization[edit]

text:
While we are disheartened that the burgeoning relationship between Callie and Erica has come to an end, the character of Callie, who has now been identified as a lesbian by show creator Shonda Rhimes, remains and her journey continues.

Gay & Lesbian Alliance Against Defamation on Torres' bisexual storyline[7]

text: The American Broadcasting Company (ABC) characterized Torres as "driven", "determined", and "outgoing", while also noting her weaknesses: "defensive" and "impulsive".[8] Ramirez described her character as someone who "appears to be a certain way, but has some very complex issues going on behind the scenes. She seems to be a very strong personality, someone who really believes in herself and has worked very hard to get where she is. She's very competitive but does have a sense of self, so she doesn't need to win all the time."[9] At Torres' initial appearance on the show, she was disliked by fans, due to her getting in the way of O'Malley and Meredith Grey (Ellen Pompeo)'s relationship. When asked of this, Ramirez said: "You do run across a lot of people who are extremely invested in that story line. Obviously, I've heard some negative stuff."[2] The end of the show's fourth season saw Torres grow closer to cardiothoracic surgeon Erica Hahn, in a relationship referred to by the portmanteau "Eri-Cal"[10] and later "Callica" by Michael Ausiello of Entertainment Weekly.[11] Rhimes stated that: "Callie and Erica have an undeniable chemistry, and watching the story unfold is something the writers are looking forward to. I wanted to illuminate their relationship in the same way we do all relationships on the show — it will be funny, sweet, honest, and a little bit dirty."[10]

text: She explained that in developing the relationship between the two: "we wanted it to be real – not some stunt to get people talking. We wanted to see what would happen if a woman suddenly had feelings for another woman."[13] The two characters shared a kiss at the end of the season four finale, with which D. Williams of After Ellen noted: "Callie and Erica became the only regular lesbian/bisexual female characters currently on network television. This is also the first time that two regular characters on a network show have begun a lesbian romance, as opposed to one becoming involved with a new lesbian character introduced expressly for that relationship."[14] Before continuing with the storyline, the show's producers consulted with the Gay & Lesbian Alliance Against Defamation to ensure they maintained realism throughout.[14] Trish Doolan, star of April's Shower was invited to consult in the workshop sessions which took place, and surmised that; "They were really wanting to be truthful to the two characters they're focusing on in the woman-woman relationship",[14] with Nikki Weiss, who also consulted, adding: "they didn't want to stereotype anything either, and write from a place where they didn't understand it. [...] I don't think they did it as a stunt to get people back to watching after the strike. I really think that they wanted to develop these two characters, and that you could see a closeness with them way before they ever decided any kind of — I think they just have a chemistry together, as actresses, too. You could tell that in the room. They definitely have a chemistry."[14]

text: In the show's fifth season, Torres embarked on a relationship with Arizona Robbins (Capshaw). The relationship between the two is referred to by the portmanteau "Calzona".[15] Rhimes praised the chemistry between Arizona and Callie in contrast, comparing it to that between the show's primary couple Grey and Derek Shepherd (Patrick Dempsey),[16] and stating: "They have that little thing that makes you want to watch them."[17] In "An Honest Mistake", Torres was initially rejected by Robbins, with Robbins citing Torres' "inexperience" as a factor. Series writer Peter Nowalk offered the insight:

text:
“
I totally understand why Arizona wouldn't want to date a newborn. It's like getting a Freshman as your Physics lab partner even though you're a Senior who not only knows the Laws of Motion but has mastered them in ways that would rock that Freshman's world. Which is not to say the Freshman won't grow to be really good at Physics, or that Callie won't catch up to Arizona on the lesbian front, it's simply that Arizona might not have the patience to wait that long.
”

text: The couple ultimately decided to begin a relationship, however, the two experience difficulties, but eventually reunite. Rhimes commented on their reconciliation: "I love [Callie] with Arizona. [...] I like that they make me feel hopeful about love."[18] Rhimes has mentioned of their relationship in the sixth season: "I would like to see Callie happily in a long-term relationship. We have so much to explore with them, because we barely know anything about [Arizona]."[19]

text: Callie and Arizona have a five-year relationship, ultimately marrying in the seventh season and divorcing in the eleventh season. Callie starts a new relationship with Penny Blake and leaves to go to New York with her in the twelfth-season finale. Shonda Rhimes spoke on Ramirez's abrupt departure, saying, "This one was different because it wasn’t a big planned thing. I had a different plan going and when Sara came in and said, ‘I really need to take this break,’ I was lucky that we’d shot the end of the season with her going to New York.”[20]

text: Reception[edit]

text: "What was truly revolutionary about Callie is that she was never defined by any single aspect of herself. She was curvy, Latina, and bisexual, and she was also a funny, frustrating, and lovable doctor with an amazing career and strong friendships. Callie was never expected to be perfect. She was allowed to be as messy as every other character on Grey's Anatomy, and, for everyone who identified with her as a whole or in part, it was a huge relief to see this woman who represented so many perspectives be allowed to just be awesome." —Sabienna Bowmna of Bustle[21]

text: Maureen Ryan from the Chicago Tribune was critical about the character's initial development. Reviewing the third season's premiere, she wrote the writers should give up on Callie explaining they have made her "far too obviously the “weird girl,” but there’s nothing underneath her brusque persona."[22] Her marriage to O'Malley was well received; Staci Krause of IGN wrote: "Their relationship has been a roller coaster and it was nice to see this turn of events, as she really is a perfect contrast to George."[23] Discussing the character in terms of her relationship with Hahn, Williams was largely positive, assessing that: "The story line offered both the drama Grey's is known for and a truthfulness network television has rarely achieved when it comes to lesbian relationships."[14] Trish Doolan and Nikki Weiss, invited by GLAAD to consult with Grey's Anatomy producers on the storyline, praised the effort put into researching the issue by the writers and actors involved, though were more negative on the scene which saw Hahn kiss Callie in an elevator in front of Mark Sloan. Weiss commented: "I just felt like, if they really cared about each other, I don't think they would do that as a stunt. That seemed a little, I don't know, forced. [...] [It] was more like a conquest, like he could have [Hahn] too or something."[14] LGBT website AfterEllen.com agreed with this view, criticizing the way the scene was edited so as to keep cutting to Sloan's point of view, as though "privileging the male gaze."[14] AfterEllen.com included Torres in their list of the Top 50 Lesbian and Bisexual Characters, ranking her at No. 6[24] and in their Top 50 Favorite Female TV Characters.[25] The character was also listed in Wetpaint's "10 Hottest TV Doctors on TV".[26]

text: Commenting on Hahn's abrupt departure from the show, Dorothy Snarker, writing for AfterEllen.com, observed of Torres and Robbins' relationship: "I [...] can't help but be wary of how the Grey's writers will handle this relationship. Jessica has proven lovely and likable in her brief screen time so far. But it's not how the romance starts, but what happens next that really matters."[15] Speaking of the musical episode, Nicole Golden of TV Fanatic called Ramirez's rendition of "Chasing Cars" "amazing" and referred to her performance in Kate Havnevik's "Grace" as "beautiful".[27] Margaret Lyons of New York Magazine was happy with the evolution of Bailey and Torres' friendship in the first half of the ninth season, calling it "the one bright spot": "They joke, they tease each other, they offer sage love advice to one another, now that they're both on their second marriages."[28] Ramirez was nominated for Outstanding Actress in a Drama Television Series at the Alma Awards, in 2007 and 2008.[29] Also in 2007, at the 13th Screen Actors Guild Awards, Ramirez and the cast of Grey's Anatomy, were the recipients of the Award for Outstanding Performance by an Ensemble in a Drama Series.[30] She and the cast were nominated for the same award, in 2008.[31] In 2011, at the 42nd NAACP Image Awards, Ramirez was nominated for Outstanding Supporting Actress in a Drama Series.[32] She was ranked at no. 7 in AfterEllen.com's list of "Top 50 Favorite Female TV Characters"[33]


document_title: Quadratic equation

question_text: what is another word for the solution of a quadratic equation

question_tokens: what is another word for the solution of a quadratic equation

text: Quadratic equation

text: Solving the quadratic equation[edit]

text: A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.

text: Factoring by inspection[edit]

text: It may be possible to express a quadratic equation ax2 + bx + c = 0 as a product (px + q)(rx + s) = 0. In some cases, it is possible, by simple inspection, to determine values of p, q, r, and s that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if px + q = 0 or rx + s = 0. Solving these two linear equations provides the roots of the quadratic.

text: For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed.[2]:202–207 If one is given a quadratic equation in the form x2 + bx + c = 0, the sought factorization has the form (x + q)(x + s), and one has to find two numbers q and s that add up to b and whose product is c (this is sometimes called "Vieta's rule"[3] and is related to Vieta's formulas). As an example, x2 + 5x + 6 factors as (x + 3)(x + 2). The more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.

text: Except for special cases such as where b = 0 or c = 0, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.[2]:207

text: Completing the square[edit]

text: The process of completing the square makes use of the algebraic identity

text: which represents a well-defined algorithm that can be used to solve any quadratic equation.[2]:207 Starting with a quadratic equation in standard form, ax2 + bx + c = 0

text:
Divide each side by a, the coefficient of the squared term.
Subtract the constant term c/a from both sides.
Add the square of one-half of b/a, the coefficient of x, to both sides. This "completes the square", converting the left side into a perfect square.
Write the left side as a square and simplify the right side if necessary.
Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side.
Solve the two linear equations.

tag: ol

text: We illustrate use of this algorithm by solving 2x2 + 4x − 4 = 0

text: The plus-minus symbol "±" indicates that both x = −1 + √3 and x = −1 − √3 are solutions of the quadratic equation.[4]

text: Quadratic formula and its derivation[edit]

text: Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula.[5] The mathematical proof will now be briefly summarized.[6] It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:

text: Taking the square root of both sides, and isolating x, gives:

text: Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax2 + 2bx + c = 0 or ax2 − 2bx + c = 0 ,[7] where b has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent.

text: A number of alternative derivations can be found in the literature. These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.

text: A lesser known quadratic formula, as used in Muller's method, and which can be found from Vieta's formulas, provides the same roots via the equation:

text: One property of this form is that it yields one valid root when a = 0, while the other root contains division by zero, because when a = 0, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an indeterminate form 0/0 for the other root. On the other hand, when c = 0, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form 0/0.

text: Reduced quadratic equation[edit]

text: It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by a, which is always possible since a is non-zero. This produces the reduced quadratic equation:[8]

text: where p = b/a and q = c/a. This monic equation has the same solutions as the original.

text: The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is:

text: or equivalently:

text: Discriminant[edit]

text: In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case D or an upper case Greek delta:[9]

text: A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:

text: If the discriminant is positive, then there are two distinct roots

text: If the discriminant is zero, then there is exactly one real root

text: If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots[10]

text: Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.

text: Geometric interpretation[edit]

text:
Roots and y-intercept in red
Vertex and axis of symmetry in blue
Focus and directrix in pink

text:
The function f(x) = ax2 + bx + c is the quadratic function.[12] The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of a, b, and c. As shown in Figure 1, if a > 0, the parabola has a minimum point and opens upward. If a < 0, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The x-coordinate of the vertex will be located at

x
=

−
b

2
a

{\displaystyle \scriptstyle x={\tfrac {-b}{2a}}}

, and the y-coordinate of the vertex may be found by substituting this x-value into the function. The y-intercept is located at the point (0, c).

text: The solutions of the quadratic equation ax2 + bx + c = 0 correspond to the roots of the function f(x) = ax2 + bx + c, since they are the values of x for which f(x) = 0. As shown in Figure 2, if a, b, and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x-coordinates of the points where the graph touches the x-axis. As shown in Figure 3, if the discriminant is positive, the graph touches the x-axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the x-axis.

text: Quadratic factorization[edit]

text: The term

text: is a factor of the polynomial

text: if and only if r is a root of the quadratic equation

text: It follows from the quadratic formula that

text: In the special case b2 = 4ac where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as

text: Graphing for real roots[edit]

text: For most of the 20th century, graphing was rarely mentioned as a method for solving quadratic equations in high school or college algebra texts. Students learned to solve quadratic equations by factoring, completing the square, and applying the quadratic formula. Recently, graphing calculators have become common in schools and graphical methods have started to appear in textbooks, but they are generally not highly emphasized.[13]

text: Being able to use a graphing calculator to solve a quadratic equation requires the ability to produce a graph of y = f(x), the ability to scale the graph appropriately to the dimensions of the graphing surface, and the recognition that when f(x) = 0, x is a solution to the equation. The skills required to solve a quadratic equation on a calculator are in fact applicable to finding the real roots of any arbitrary function.

text: Since an arbitrary function may cross the x-axis at multiple points, graphing calculators generally require one to identify the desired root by positioning a cursor at a "guessed" value for the root. (Some graphing calculators require bracketing the root on both sides of the zero.) The calculator then proceeds, by an iterative algorithm, to refine the estimated position of the root to the limit of calculator accuracy.

text: Avoiding loss of significance[edit]

text: Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis, where real numbers are approximated by floating point numbers (called "reals" in many programming languages). In this context, the quadratic formula is not completely stable.

text:
This occurs when the roots have different order of magnitude, or, equivalently, when b2 and b2 − 4ac are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root. To avoid this, the root that is smaller in magnitude, r, can be computed as

(
c

/

a
)

/

R

{\displaystyle (c/a)/R}

 where R is the root that is bigger in magnitude.

text: A second form of cancellation can occur between the terms b2 and 4ac of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots.[7][14]

text: Examples and applications[edit]

text:
The golden ratio is found as the solution of the quadratic equation

x

2

−
x
−
1
=
0.

{\displaystyle x^{2}-x-1=0.}

text: The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas—are quadratic equations in two variables.

text: Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large involves solving a quadratic equation.

text: The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation.

text: Descartes' theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation.

text: The equation given by Fuss' theorem, giving the relation among the radius of a bicentric quadrilateral's inscribed circle, the radius of its circumscribed circle, and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the excircle of an ex-tangential quadrilateral.

text: History[edit]

text: Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian clay tablets) could solve problems relating the areas and sides of rectangles. There is evidence dating this algorithm as far back as the Third Dynasty of Ur.[15] In modern notation, the problems typically involved solving a pair of simultaneous equations of the form:

text: which is equivalent to the statement that x and y are the roots of the equation:[16]:86

text: The steps given by Babylonian scribes for solving the above rectangle problem, in terms of x and y, were as follows:

text:
Compute half of p.
Square the result.
Subtract q.
Find the (positive) square root using a table of squares.
Add together the results of steps (1) and (4) to give x. In modern notation this means calculating

x
=

p
2

+

(

p
2

)

2

−
q

.

{\displaystyle x={\frac {p}{2}}+{\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}.}

text: Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation.[17] Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots.[18][19] Rules for quadratic equations were given in The Nine Chapters on the Mathematical Art, a Chinese treatise on mathematics.[19][20] These early geometric methods do not appear to have had a general formula. Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC. With a purely geometric approach Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation. In his work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive.[21]

text: In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation ax2 + bx = c as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value." (Brahmasphutasiddhanta, Colebrook translation, 1817, page 346)[16]:87 This is equivalent to:

text: The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ax/c = y[clarification needed : this is linear, not quadratic]). Muhammad ibn Musa al-Khwarizmi (Persia, 9th century), inspired by Brahmagupta,[original research?] developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process.[22] He also described the method of completing the square and recognized that the discriminant must be positive,[22][23]:230 which was proven by his contemporary 'Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution.[23]:234 While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions,[22]:191 as well as irrational numbers as solutions.[24] Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.[25] The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations.[26]

text: The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation.[27] His solution was largely based on Al-Khwarizmi's work.[22] The writing of the Chinese mathematician Yang Hui (1238–1298 AD) is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.[28] By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594.[29] In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today. The first appearance of the general solution in the modern mathematical literature appeared in an 1896 paper by Henry Heaton.[30]

text: Advanced topics[edit]

text: Alternative methods of root calculation[edit]

text: Vieta's formulas[edit]

text: Vieta's formulas give a simple relation between the roots of a polynomial and its coefficients. In the case of the quadratic polynomial, they take the following form:

text: and

text: These results follow immediately from the relation:

text: which can be compared term by term with

text: The first formula above yields a convenient expression when graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, when there are two real roots the vertex's x-coordinate is located at the average of the roots (or intercepts). Thus the x-coordinate of the vertex is given by the expression

text: The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving

text: As a practical matter, Vieta's formulas provide a useful method for finding the roots of a quadratic in the case where one root is much smaller than the other. If | x 2| << | x 1|, then x 1 + x 2 ≈ x 1, and we have the estimate:

text: The second Vieta's formula then provides:

text: These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large b), which causes round-off error in a numerical evaluation. Figure 5 shows the difference between (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient b increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as b increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse.

text: This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see step response).

text: Trigonometric solution[edit]

text: In the days before calculators, people would use mathematical tables—lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. Methods of numerical approximation existed, called prosthaphaeresis, that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots.[13] Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in celestial mechanics calculations.

text: It is within this context that we may understand the development of means of solving quadratic equations by the aid of trigonometric substitution. Consider the following alternate form of the quadratic equation,

text:
[1]

a

x

2

+
b
x
±
c
=
0
,

{\displaystyle ax^{2}+bx\pm c=0,}

text: where the sign of the ± symbol is chosen so that a and c may both be positive. By substituting

text:
[2]

x
=

c

/

a

tan

θ

{\displaystyle x={\sqrt {c/a}}\tan \theta }

text: and then multiplying through by cos2θ, we obtain

text:
[3]

sin

2

θ
+

b

a
c

sin

θ
cos

θ
±

cos

2

θ
=
0.

{\displaystyle \sin ^{2}\theta +{\frac {b}{\sqrt {ac}}}\sin \theta \cos \theta \pm \cos ^{2}\theta =0.}

text: Introducing functions of 2θ and rearranging, we obtain

text:
[4]

tan

2

θ

n

=
+
2

a
c

b

,

{\displaystyle \tan 2\theta _{n}=+2{\frac {\sqrt {ac}}{b}},}

text:
[5]

sin

2

θ

p

=
−
2

a
c

b

,

{\displaystyle \sin 2\theta _{p}=-2{\frac {\sqrt {ac}}{b}},}

text: where the subscripts n and p correspond, respectively, to the use of a negative or positive sign in equation [1]. Substituting the two values of θn or θp found from equations [4] or [5] into [2] gives the required roots of [1]. Complex roots occur in the solution based on equation [5] if the absolute value of sin 2θp exceeds unity. The amount of effort involved in solving quadratic equations using this mixed trigonometric and logarithmic table look-up strategy was two-thirds the effort using logarithmic tables alone.[31] Calculating complex roots would require using a different trigonometric form.[32]

text:
A seven-place lookup table might have only 100,000 entries, and computing intermediate results to seven places would generally require interpolation between adjacent entries.

log

a
=
0.6192290
,
log

b
=
0.9618637
,
log

c
=
1.0576927

{\displaystyle \log a=0.6192290,\log b=0.9618637,\log c=1.0576927}

2

a
c

/

b
=
2
×

10

(
0.6192290
+
1.0576927
)

/

2
−
0.9618637

=
1.505314

{\displaystyle 2{\sqrt {ac}}/b=2\times 10^{(0.6192290+1.0576927)/2-0.9618637}=1.505314}

θ
=
(

tan

−
1

1.505314
)

/

2
=

28.20169

∘

 or

−

61.79831

∘

{\displaystyle \theta =(\tan ^{-1}1.505314)/2=28.20169^{\circ }{\text{ or }}-61.79831^{\circ }}

log

|

tan

θ

|

=
−
0.2706462

 or

0.2706462

{\displaystyle \log |\tan \theta |=-0.2706462{\text{ or }}0.2706462}

log

c

/

a

=
(
1.0576927
−
0.6192290
)

/

2
=
0.2192318

{\displaystyle \log {\sqrt {c/a}}=(1.0576927-0.6192290)/2=0.2192318}

x

1

=

10

0.2192318
−
0.2706462

=
0.888353

{\displaystyle x_{1}=10^{0.2192318-0.2706462}=0.888353}

 (rounded to six significant figures)

text: Solution for complex roots in polar coordinates[edit]

text:
If the quadratic equation

a

x

2

+
b
x
+
c
=
0

{\displaystyle ax^{2}+bx+c=0}

 with real coefficients has two complex roots—the case where

b

2

−
4
a
c
<
0
,

{\displaystyle b^{2}-4ac<0,}

 requiring a and c to have the same sign as each other—then the solutions for the roots can be expressed in polar form as[33]

text:
where

r
=

c
a

{\displaystyle r={\sqrt {\tfrac {c}{a}}}}

 and

θ
=

cos

−
1

(

−
b

2

a
c

)

.

{\displaystyle \theta =\cos ^{-1}\left({\tfrac {-b}{2{\sqrt {ac}}}}\right).}

text: Geometric solution[edit]

text: The quadratic equation may be solved geometrically in a number of ways. One way is via Lill's method. The three coefficients a, b, c are drawn with right angles between them as in SA, AB, and BC in Figure 6. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient a or SA. If a is 1 the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA.[34]

text: The Carlyle circle, named after Thomas Carlyle, has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis.[35] Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.

text: Generalization of quadratic equation[edit]

text: The formula and its derivation remain correct if the coefficients a, b and c are complex numbers, or more generally members of any field whose characteristic is not 2. (In a field of characteristic 2, the element 2a is zero and it is impossible to divide by it.)

text: The symbol

text: in the formula should be understood as "either of the two elements whose square is b2 − 4ac, if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2. Even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field.

text: Characteristic 2[edit]

text: In a field of characteristic 2, the quadratic formula, which relies on 2 being a unit, does not hold. Consider the monic quadratic polynomial

text: over a field of characteristic 2. If b = 0, then the solution reduces to extracting a square root, so the solution is

text: and there is only one root since

text: In summary,

text: See quadratic residue for more information about extracting square roots in finite fields.

text: In the case that b ≠ 0, there are two distinct roots, but if the polynomial is irreducible, they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the 2-root R(c) of c to be a root of the polynomial x2 + x + c, an element of the splitting field of that polynomial. One verifies that R(c) + 1 is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic ax2 + bx + c are

text: and

text: For example, let a denote a multiplicative generator of the group of units of F4, the Galois field of order four (thus a and a + 1 are roots of x2 + x + 1 over F4. Because (a + 1)2 = a, a + 1 is the unique solution of the quadratic equation x2 + a = 0. On the other hand, the polynomial x2 + ax + 1 is irreducible over F4, but it splits over F16, where it has the two roots ab and ab + a, where b is a root of x2 + x + a in F16.

text: This is a special case of Artin–Schreier theory.