To determine the values of [tex]\( x \)[/tex] that are not in the domain of the function [tex]\( g(x) = \frac{x-2}{x^2 + 10x + 24} \)[/tex], we need to identify where the denominator is equal to zero, since division by zero is undefined.
Start by examining the denominator of [tex]\( g(x) \)[/tex]:
[tex]\[ x^2 + 10x + 24 \][/tex]
We seek the values of [tex]\( x \)[/tex] for which this quadratic expression equals zero:
[tex]\[ x^2 + 10x + 24 = 0 \][/tex]
This is a standard quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. To solve it, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = 24 \)[/tex]. Substitute these values into the quadratic formula:
[tex]\[ x = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} \][/tex]
First, compute the discriminant ([tex]\( b^2 - 4ac \)[/tex]):
[tex]\[ \text{Discriminant} = 10^2 - 4 \cdot 1 \cdot 24 = 100 - 96 = 4 \][/tex]
Now, use the discriminant to find the roots of the quadratic equation:
[tex]\[ x = \frac{-10 \pm \sqrt{4}}{2} \][/tex]
[tex]\[ x = \frac{-10 \pm 2}{2} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{-10 + 2}{2} = \frac{-8}{2} = -4 \][/tex]
[tex]\[ x_2 = \frac{-10 - 2}{2} = \frac{-12}{2} = -6 \][/tex]
Thus, the values of [tex]\( x \)[/tex] that make the denominator zero and are therefore not in the domain of [tex]\( g(x) \)[/tex] are [tex]\( -4 \)[/tex] and [tex]\( -6 \)[/tex].
Therefore, the values of [tex]\( x \)[/tex] that are NOT in the domain of [tex]\( g \)[/tex] are:
[tex]\[ x = -4, -6 \][/tex]
