已知方程 $3x^2 + 9x - k = 0$ 的两个根之差为 13 个单位，其中 $k$ 为常数。求 $k$ 的值。 答：$k$ = ...

1. Express the sum and product of the roots in terms of $k$.

Let the roots of the equation be $\alpha$ and $\beta$.
Sum of roots, $\alpha + \beta = -\frac{9}{3} = -3$
Product of roots, $\alpha\beta = -\frac{k}{3}$

2. Use the algebraic identity relating the difference, sum, and product of roots.

We are given that the difference of the roots is 13, so $|\alpha - \beta| = 13$.
Using the identity $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$:
$13^2 = (-3)^2 - 4\left(-\frac{k}{3}\right)$

3. Solve the resulting equation for $k$.

$169 = 9 + \frac{4k}{3}$
$\frac{4k}{3} = 160$
$4k = 480$
$k = 120$