To find the [tex]$x$[/tex]-intercepts of the graph of the function [tex]\( f(x) = x^2 + 4x - 12 \)[/tex], we need to determine the points where the graph intersects the x-axis. At the x-intercepts, the value of [tex]\( y \)[/tex] (or [tex]\( f(x) \)[/tex]) is zero. Therefore, we need to solve the equation:
[tex]\[ x^2 + 4x - 12 = 0 \][/tex]
This is a quadratic equation, so we can solve for [tex]\( x \)[/tex] using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = -12 \)[/tex]
Plugging the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1} \][/tex]
Calculate the discriminant:
[tex]\[ 4^2 - 4 \cdot 1 \cdot (-12) = 16 + 48 = 64 \][/tex]
So, the formula now becomes:
[tex]\[ x = \frac{-4 \pm \sqrt{64}}{2} \][/tex]
Simplify further:
[tex]\[ x = \frac{-4 \pm 8}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{-4 + 8}{2} = \frac{4}{2} = 2 \][/tex]
and
[tex]\[ x = \frac{-4 - 8}{2} = \frac{-12}{2} = -6 \][/tex]
Thus, the solutions to the equation are [tex]\( x = 2 \)[/tex] and [tex]\( x = -6 \)[/tex]. These solutions represent the x-intercepts of the graph of the function [tex]\( f(x) \)[/tex].
Therefore, the x-intercepts are at the points [tex]\( (-6, 0) \)[/tex] and [tex]\( (2, 0) \)[/tex].
So, the correct answer is:
[tex]\[ (-6, 0),(2, 0) \][/tex]
