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Given a general quadratic equation of the form + + = , with representing an unknown, and coefficients , , and representing known real or complex numbers with , the values of satisfying the equation, called the roots or zeros, can be found using the quadratic formula,
He presented a method of completing the square to solve quadratic equations, sometimes called Śrīdhara's method or the Hindu method. Because the quadratic formula can be derived by completing the square for a generic quadratic equation with symbolic coefficients, it is called Śrīdharācārya's formula in some places.
Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.
Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where b and c are positive integers) squares equal roots (ax 2 = bx) squares equal number (ax 2 = c) roots equal number (bx = c) squares and roots equal number (ax 2 + bx = c) squares and number equal roots (ax 2 ...
The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra , analytic geometry , and in the majority of applications of quadratic forms, the coefficients are real or complex numbers.
To solve a quadratic equation, the Babylonians essentially used the standard quadratic formula. They considered quadratic equations of the form: + = where b and c were not necessarily integers, but c was always positive. They knew that a solution to this form of equation is: [15]
It is also used for graphing quadratic functions, deriving the quadratic formula, and more generally in computations involving quadratic polynomials, for example in calculus evaluating Gaussian integrals with a linear term in the exponent, [2] and finding Laplace transforms. [3] [4]
A recent study attributes a greater role to Stevin in developing the real numbers than has been acknowledged by Weierstrass's followers. [20] Stevin proved the intermediate value theorem for polynomials, anticipating Cauchy's proof thereof. Stevin uses a divide and conquer procedure, subdividing the interval into ten equal parts. [21]