# How do you solve the quadratic using the quadratic formula given a^2-7a-10=0?

Aug 10, 2016

$a = \frac{7}{2} \pm \frac{\sqrt{89}}{2}$

#### Explanation:

The slight nuisance with this question is that the $a$ used as a variable name clashes with the $a , b , c$ that normally stand for the coefficients of the terms of a quadratic.

To sidestep this issue, consider the quadratic equation:

${x}^{2} - 7 x - 10 = 0$

This has the same zeros as our original quadratic, just expressed as values of $x$ instead of values of $a$.

Our quadratic equation in $x$ is of the form $a {x}^{2} + b x + c = 0$ with $a = 1$, $b = - 7$ and $c = - 10$.

It therefore has zeros given by the quadratic formula:

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

$= \frac{7 \pm \sqrt{{\left(- 7\right)}^{2} - 4 \left(1\right) \left(- 10\right)}}{2 \cdot 1}$

$= \frac{7 \pm \sqrt{49 + 40}}{2}$

$= \frac{7 \pm \sqrt{89}}{2}$

$= \frac{7}{2} \pm \frac{\sqrt{89}}{2}$

So the zeros of our original quadratic are given by:

$a = \frac{7}{2} \pm \frac{\sqrt{89}}{2}$