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Concept
Quadratic formula
Summary
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others.
Given a general quadratic equation of the form
whose discriminant is positive, with x representing an unknown, with a, b and c representing constants, and with a ≠ 0, the quadratic formula is:
where the plus–minus symbol "±" indicates that the quadratic equation has two solutions. Written separately, they become:
Each of these two solutions is also called a root (or zero) of the quadratic equation. Geometrically, these roots represent the x-values at which any parabola, explicitly given as y = ax2 + bx + c, crosses the x-axis.
As well as being a formula that yields the zeros of any parabola, the quadratic formula can also be used to identify the axis of symmetry of the parabola, and the number of real zeros the quadratic equation contains.
The expression b2 − 4ac is known as the discriminant. If a, b, and c are real numbers and a ≠ 0 then
When b2 − 4ac > 0, there are two distinct real roots or solutions to the equation ax^2 + bx + c = 0.
When b2 − 4ac = 0, there is one repeated real solution.
When b2 − 4ac < 0, there are two distinct complex solutions, which are complex conjugates of each other.
The quadratic formula, in the case when the discriminant is positive, may also be written as
which may be simplified to
This version of the formula makes it easy to find the roots when using a calculator.
When b is an even integer, it is usually easier to use the reduced formula
In the case when the discriminant is negative, complex roots are involved. The quadratic formula can be written as:
A lesser known quadratic formula, also named "citardauq", which is used in Muller's method and which can be found from Vieta's formulas, provides (assuming a ≠ 0, c ≠ 0) the same roots via the equation:
For positive , the subtraction causes cancellation in the standard formula (respectively negative and addition), resulting in poor accuracy.
Official source
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