Answer
The solutions to the given equation are $x = 6$ and $x = -2$.
Work Step by Step
First, rewrite the equation in quadratic form, which is $ax^2 + bx + c = 0$.
$x^2 - 4x - 12 = 0$
We need to find which factors multiplied together will equal $ac$ but when added together will equal $b$.
In this equation, $a = 1$, $b = -4$ and $c = -12$; therefore, $ac = -12$.
Looking at the equation, one of the factors would have to be positive and the other negative, but the negative factor would have the greater absolute value. Let's look at the possibilities:
$-6$ and $2$
$-4$ and $3$
$-12$ and $1$
The first option will work. Rewrite the equation in factor form:
$(x - 6)(x + 2) = 0$
According to the Zero Product Property, set the first factor equal to $0$:
$x - 6 = 0$
Add $6$ to each side of the equation:
$x = 6$
Set the second factor equal to $0$:
$x + 2 = 0$
Subtract $2$ from each side to solve for $x$:
$x = -2$
The solutions to this equation are $x = 6$ and $x = -2$.
