The difference of the two roots of the equation $3x^2 + 9x - k = 0$ is 13 units, where $k$ is a constant. Find the value of $k$. Answer k = ...

1. Write the difference between the two roots in terms of the discriminant.

For $3x^2+9x-k=0$, $a=3$, $b=9$, $c=-k$.

The difference between the two roots is
$\frac{\sqrt{b^2-4ac}}{|a|}.$
Given that the difference is $13$,
$\frac{\sqrt{9^2-4(3)(-k)}}{3}=13.$

2. Simplify the equation involving the discriminant.

$\frac{\sqrt{81+12k}}{3}=13$
$\sqrt{81+12k}=39$

3. Square both sides and solve for $k$.

$81+12k=39^2$
$81+12k=1521$
$12k=1440$
$k=120$