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

We start with


Divide by $a$:


Add $\frac{b^{2}}{4a^{2}}$ to both sides in order to complete the square:


Complete the square:


Move $\frac{c}{a}$ to the other side:


Take the square root of both sides:


Finally, move the $\frac{b}{2a}$ to the other side:


This is the quadratic formula, and we are done.


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


See Also

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