Quadratic formula
Formula that provides the solutions to a quadratic equation / From Wikipedia, the free encyclopedia
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In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions.
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,
where the plus–minus symbol "" indicates that the equation has two roots.[1] Written separately, these are:
The quantity is known as the discriminant of the quadratic equation.[2] If the coefficients , , and are real numbers then when , the equation has two distinct real roots; when , the equation has one repeated real root; and when , the equation has two distinct complex roots, which are complex conjugates of each other.
Geometrically, the roots represent the values at which the graph of the quadratic function , a parabola, crosses the -axis.[3] The quadratic formula can also be used to identify the parabola's axis of symmetry.[4]
The standard way to derive the quadratic formula is to apply the method of completing the square to the generic quadratic equation .[5][6][7][8] The idea is to manipulate the equation into the form for some expressions and written in terms of the coefficients; take the square root of both sides; and then isolate .
We start by dividing the equation by the quadratic coefficient , which is allowed because is non-zero. Afterwards, we subtract the constant term to isolate it on the right-hand side:
The left-hand side is now of the form , and we can "complete the square" by adding a constant to obtain a squared binomial
Because the left-hand side is now a perfect square, we can easily take the square root of both sides:
Finally, subtracting from both sides to isolate produces the quadratic formula:
The quadratic formula can equivalently be written using various alternative expressions, for instance
which can be derived by first dividing a quadratic equation by , resulting in , then substituting the new coefficients into the standard quadratic formula. Because this variant allows re-use of the intermediately calculated quantity , it can slightly reduce the arithmetic involved.
Square root in the denominator
A lesser known quadratic formula, first mentioned by Giulio Fagnano,[9] describes the same roots via an equation with the square root in the denominator (assuming ):
Here the minus–plus symbol "" indicates that the two roots of the quadratic equation, in the same order as the standard quadratic formula, are
This variant has been jokingly called the "citardauq" formula ("quadratic" spelled backwards).[10]
When has the opposite sign as either or , subtraction can cause catastrophic cancellation, resulting in poor accuracy in numerical calculations; choosing between the version of the quadratic formula with the square root in the numerator or denominator depending on the sign of can avoid this problem. See § Numerical calculation below.
This version of the quadratic formula is used in Muller's method for finding the roots of general functions. It can be derived from the standard formula from the identity , one of Vieta's formulas. Alternately, it can be derived by dividing the original equation , by to get , applying the standard formula to find the two roots , and then taking the reciprocal to find the roots of the original equation.