# 7.3 The Quadratic Formula

**The Quadratic Formula • Character of the Roots of a Quadratic Equation**

We now use the method of completing the square to derive a general formula that may be used for the solution of any quadratic equation.

Consider the general quadratic equation:

When we divide through by `a`, we obtain

Subtracting `c`/`a` from each side, we have

Half of `b`/`a` is `b`/2`a`, which squared is ${b}^{2}\hspace{0.17em}/\hspace{0.17em}4{a}^{2}\hspace{0.17em}.\hspace{0.17em}$ Adding ${b}^{2}\hspace{0.17em}/\hspace{0.17em}4{a}^{2}$ to each side gives us

Writing the left side as a perfect square and combining fractions on the right side,

Equating $x\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{b}{2a}}$ to the positive and negative square root of the right side,

When ...

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