# How do you solve by using the quadratic formula: x^2 + 7x = -3?

May 9, 2016

$x = - \frac{7}{2} \pm \frac{\sqrt{37}}{2}$

#### Explanation:

First add $3$ to both sides to get:

${x}^{2} + 7 x + 3 = 0$

This is in the form $a {x}^{2} + b x + c = 0$ with $a = 1$, $b = 7$ and $c = 3$.

The roots are given by the quadratic formula:

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

$= \frac{- 7 \pm \sqrt{{7}^{2} - \left(4 \cdot 1 \cdot 3\right)}}{2 \cdot 1}$

$= \frac{- 7 \pm \sqrt{49 - 12}}{2}$

$= \frac{- 7 \pm \sqrt{37}}{2}$

$= - \frac{7}{2} \pm \frac{\sqrt{37}}{2}$

Note that since $37$ is prime, the square root does not simplify further.

$\sqrt{37}$ does have a simple continued fraction expansion which you can truncate to give rational approximations:

sqrt(37) = [6;bar(12)] = 6+1/(12+1/(12+1/(12+1/(12+1/(12+...)))))