Question

solve the equation for all real solutions in simplest form.
$3m^2 + 12m + 13 = -2m$

Explanation:

Step1: Rearrange into standard quadratic form

$3m^2 + 12m + 2m + 13 = 0$
$3m^2 + 14m + 13 = 0$

Step2: Apply quadratic formula

For $ax^2+bx+c=0$, $m=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Here $a=3$, $b=14$, $c=13$
$$m=\frac{-14\pm\sqrt{14^2-4(3)(13)}}{2(3)}$$

Step3: Calculate discriminant

$\sqrt{196 - 156}=\sqrt{40}=2\sqrt{10}$
$$m=\frac{-14\pm2\sqrt{10}}{6}$$

Step4: Simplify the fraction

$m=\frac{-7\pm\sqrt{10}}{3}$

Answer:

$m=\frac{-7+\sqrt{10}}{3}$ and $m=\frac{-7-\sqrt{10}}{3}$