Answer
$x = 12$
Work Step by Step
In this problem, we are asked to solve for $x$. First, we want to factor this quadratic equation, if possible:
$x^2 - 24x + 144 = 0$
To factor a quadratic polynomial in the form $ax^2 + bx + c$, we look for factors of $c$ that when added together equals $b$.
For the trinomial $x^2 - 24x + 144$, $c=144$ so look for factors of $144$ whose sum is $-24$. We need both factors to be negative because two negative numbers yield a positive number, but will yield a negative number when added together.
We have the following possibilities:
$144=(-144)(-1)$
$-144+(-1) = -145$
$144= (-72)(-2)$
$-72+(-2) = -74$
$144= (-12)(-12)$
$-12+(-12) = -24$
The third pair, $-12$ and $-12$, is the one we are looking for.
Thus, the factored form of the trinomial is $(x - 12)(x - 12)$ so the equation above is equivalent to:
$$(x-12)(x-12)=0$$
According to the Zero-Product Property, if the product of two factors $a$ and $b$ equals zero, then either $a$ is zero, $b$ is zero, or both equal zero. Since the two factors are the same, we just set the factor equal to $0$:
$x - 12 = 0$
Add $12$ to each side to solve for $x$:
$x = 12$
