To find the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = x^2 + 4x - 12 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. The [tex]\( x \)[/tex]-intercepts occur where the graph of the function crosses the [tex]\( x \)[/tex]-axis.
Let's set the function equal to zero:
[tex]\[
x^2 + 4x - 12 = 0
\][/tex]
This is a quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -12 \)[/tex].
To find the [tex]\( x \)[/tex]-values that satisfy this equation, we use the quadratic formula:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
First, we calculate the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 4^2 - 4 \cdot 1 \cdot (-12)
\][/tex]
[tex]\[
\Delta = 16 + 48
\][/tex]
[tex]\[
\Delta = 64
\][/tex]
Next, we substitute the discriminant and the coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex] back into the quadratic formula:
[tex]\[
x = \frac{{-4 \pm \sqrt{64}}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{{-4 \pm 8}}{2}
\][/tex]
This gives us two solutions:
[tex]\[
x_1 = \frac{{-4 + 8}}{2} = \frac{4}{2} = 2
\][/tex]
[tex]\[
x_2 = \frac{{-4 - 8}}{2} = \frac{-12}{2} = -6
\][/tex]
Thus, the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = x^2 + 4x - 12 \)[/tex] are:
[tex]\[ (2, 0) \quad \text{and} \quad (-6, 0) \][/tex]
Therefore, the correct answer from the provided options is:
[tex]\[ \boxed{(-6,0),(2,0)} \][/tex]
