Quadratic formula

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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 $ax^2+bx+c=0$.

Moving c to the other side, we obtain

$ax^2+bx=-c$

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

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

Completing the square on 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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