The problem is to solve the equation $\frac{x^2}{2} - x = 4$ for $x$.

1. Problem Description

The problem is to solve the equation $\frac{x^2}{2} - x = 4$ for $x$.

2. Solution Steps

First, we multiply both sides of the equation by 2 to eliminate the fraction:

$2(\frac{x^2}{2} - x) = 2(4)$

$x^2 - 2x = 8$

Next, we move the constant term to the left side to set the equation to zero:

$x^2 - 2x - 8 = 0$

Now, we can solve the quadratic equation by factoring. We look for two numbers that multiply to -8 and add to -

2. Those numbers are -4 and

2. Therefore, we can factor the equation as:

$(x - 4)(x + 2) = 0$

Finally, we set each factor equal to zero and solve for $x$:

$x - 4 = 0$ or $x + 2 = 0$

$x = 4$ or $x = -2$

3. Final Answer

$x = 4, -2$