Answer
The $x$-value of the $x$-intercept is $-2$.
Work Step by Step
The definition of the $x$-intercept is the value of $x$ when $y$ equals $0$.
To find the $x$ intercept of the graph given, we set $y$ or $f(x)$ equal to $0$:
$0 = x^2 + 4x + 4$
To solve for $x$, we try to factor first:
To factor a quadratic polynomial in the form $x^2 + bx + c$, we look at factors of $c$ such that, when added together, equal $b$.
For the trinomial $x^2 + 4x + 4$, $c=4$, so look for factors of $4$ that when added together will equal $b$ or $4$. Both factors must be positive.
Here are the possibilities:
$4=(4)(1)$
$4+1 = 5$
$4=(2)(2)$
$2+2 = 4$
The second pair, $2$ and $2$, is what we are looking for.
Hence, the factored of the trinomial is $(x + 2)(x + 2)$.
Thus, the equation above is equivalent to:
$$(x+2)(x+2)=0$$
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. Therefore, we can set each factor equal to zero; however, since both factors are the same, we solve for $x$ only once:
The first factor:
$x + 2 = 0$
Subtract $2$ from each side to solve for $x$:
$x = -2$
This is the $x$-value of the $x$-intercept.
