Problem 15

Explanation

Given the matrix determinant equation $\begin{vmatrix} x^2 & x \\ 2 & 1 \end{vmatrix} = 8$.

Recall that the determinant of a $2 \times 2$ matrix is the difference between the product of the main diagonal and the secondary diagonal: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$.

So, we expand the equation:

Move all terms to the left side to form a quadratic equation:

Factor the quadratic equation. We look for two numbers that multiply to $-8$ and add up to $-2$. These numbers are $-4$ and $2$.

The zeros are:

So, the possible values of $x$ are $4$ or $-2$.