# How do you solve 7a^2+24=-29a?

##### 1 Answer
Jun 18, 2015

I solve by writing as a quadratic, then factoring. (If factoring hadn't worked quickly, I would have used the quadratic formula.)

#### Explanation:

$7 {a}^{2} + 24 = - 29 a$ if and only if

$7 {a}^{2} + 29 a + 24 = 0$

If it is easily factorable, the factors must look like:

$\left(7 a + \textcolor{w h i t e}{\text{sss" )(a + color(white)"sss}}\right)$

And the spaces are filled by two factors of 24:

$\left(7 a + m\right) \left(a + n\right)$

$m \times n = 24$ where $m a + 7 n a = 29 a$

$1 \times 24$ and $24 \times 1$ won't work
$2 \times 12$ and $12 \times 2$ won't work
$3 \times 8$ won't work in that order, but $8 \times 3$ works.

Check to be sure
$\left(7 a + 8\right) \left(a + 3\right) = 7 {a}^{2} + 21 a + 8 a + 24 = 7 {a}^{2} + 29 a + 24$

So we have:
$7 {a}^{2} + 29 a + 24 = 0$

$\left(7 a + 8\right) \left(a + 3\right) = 0$

$7 a + 8 = 0$ $\text{ or }$ $a + 3 = 0$

$a = - \frac{8}{7}$ $\text{ or }$ $a = - 3$