We are asked to solve the equation $\frac{x^2}{6} - x = 12$ for $x$.

1. Problem Description

We are asked to solve the equation $\frac{x^2}{6} - x = 12$ for $x$.

2. Solution Steps

First, we want to eliminate the fraction by multiplying both sides of the equation by 6:

$6(\frac{x^2}{6} - x) = 6(12)$

$x^2 - 6x = 72$

Next, we want to set the equation equal to zero to use the quadratic formula. Subtract 72 from both sides:

$x^2 - 6x - 72 = 0$

We can use the quadratic formula to solve for $x$. The quadratic formula is:

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

In our case, $a = 1$, $b = -6$, and $c = -72$. Plugging these values into the quadratic formula:

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-72)}}{2(1)}$

$x = \frac{6 \pm \sqrt{36 + 288}}{2}$

$x = \frac{6 \pm \sqrt{324}}{2}$

$x = \frac{6 \pm 18}{2}$

Now we have two possible solutions for $x$:

$x_1 = \frac{6 + 18}{2} = \frac{24}{2} = 12$

$x_2 = \frac{6 - 18}{2} = \frac{-12}{2} = -6$

3. Final Answer

$x = 12, -6$