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Quadratic formula

From AoPSWiki

The quadratic formula is a general expression for the solutions to a quadratic equation. It is used when other methods, such as completing the square, factoring, and square root property do not work or are too tedious.

General Solution For A Quadratic by Completing the Square

Let the quadratic be in the form a\cdot x^2+b\cdot x+c=0.

Moving c to the other side, we obtain

a\cdot x^2+b\cdot x=-c

Dividing by {a} and adding \frac{b^2}{4a^2} to both sides yields

x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}.

Factoring the LHS gives

\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}

As described above, an equation in this form can be solved, yielding

{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}

This formula is also called the quadratic formula.

Given the values {a},{b},{c}, we can find all real and complex solutions to the quadratic equation.

Variation

In some situations, it is preferable to use this variation of the quadratic formula:

\frac{2c}{-b\pm\sqrt{b^2-4ac}}

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