Quadratic Equation Quadratic Formula Derivation Generalizations

The numbers a, b and c are called coefficients: a is the coefficient of x², b is the coefficient of x, and c is the free term or constant.

Take, for example, 5x² + 3x + 4 = 0. In this example, 5 is the coefficient of , 3 is the coefficient of x, and 4 is the free term.

A quadratic equation with real or complex coefficients has two complex roots (i.e., solutions) usually denoted as and , although the two roots may be equal. These roots can be computed using the quadratic formula.

Note that the highest exponent is twice the value of the exponent of the middle term. This equation may be resolved directly or with a simple substitution, using the methods that are available for the quadratic, such as factoring (also called factorising), the quadratic formula, or completing the square.

1 Quadratic formula

The quadratic formula explicitly gives the solutions of a quadratic equation in terms of the coefficients a, b and c, which we temporarily assume to be real (but see below for generalizations) with a being non-zero. These solutions are also called the roots of the equation. The formula reads

The term b² − 4ac is called the discriminant of the quadratic equation, because it discriminates between three qualitatively different cases:

Note that when computing roots numerically, the usual form of the quadratic formula is not ideal. See Loss of significance for details.

2 Derivation

The equation is now in a form in which we can conveniently complete the square. To "complete the square" is to add a constant (i.e., in this case, a quantity that does not depend on x) to the expression to the left of "=", that will make it a perfect square trinomial of the form x² + 2xy + y². Since "2xy" in this case is (b/a)x, we must have y = b/(2a), so we add the square of b/(2a) to both sides, getting

The left side is now a perfect square; it is the square of (x + b/(2a)). The right side can be written as a single fraction; the common denominator is 4a². We get

3 Generalizations

The formula and its proof remain correct if the coefficients a, b and c are complex numbers, or more generally members of any field whose characteristicThe word characteristic has several meanings: In mathematics, see characteristic (algebra) characteristic function characteristic subgroup Euler characteristic method of characteristics In genetics, see characteristic (genetics). Characteristic is also so is not 2. (In a field of characteristic 2, the element 2a is zero and it is impossible to divide by it.)

The symbol

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.

4 Viète's formulas

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

5 History

The ancient BabyloniaBabylonia was an ancient state in Mesopotamia (in modern Iraq), combining the territories of Sumer and Akkad. Its capital was Babylon. The earliest mention of Babylon can be found in a tablet of the reign of Sargon of Akkad, dating back to the 23rd centurns (around 400 BC) and ChineseThis article is on the geographic and cultural entity. For other meanings, see China (disambiguation). China ( Traditional Chinese: , Simplified Chinese: , Hanyu Pinyin: Zhongguo, Wade-Giles: Chung-kuo) is a country in continental East Asia with some oute used the method of completing the square to solve quadratic equations with positive roots, but did not have a general formula. EuclidEuclid of Alexandria ( Greek: Eukleides (circa 365 275 BC) was a Greek mathematician, now known as "the father of geometry". His most famous work is the Elements widely considered to be history's most successful textbook. Within it, the properties of geom produced a more abstract geometrical method around 300 BC.

The first mathematician known to have used the general algebraic formula, allowing negative as well as positive solutions, was Brahmagupta ( India, 7th century). Al-Khwarizmi ( Arabia, 11th century) independently developed a set of formulae that worked for positive solutions. Abraham bar Hiyya Ha-Nasi (also known by the Latin name Savasorda) was the first to introduce the complete solution to Europe in his book Liber embadorum.

Sridhara was said to be one of the first mathematicians to give a general rule for solving a quadratic equation. But there has been a dispute over his time. The rule is (as quoted by Bhaskara II):

Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root.[1]

6 Related topics

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