To determine the value of [tex]\( g \)[/tex] that satisfies the equation [tex]\( f(g) = g^2 + 3g = 18 \)[/tex], let's follow these steps:
1. Write down the given equation:
[tex]\[
g^2 + 3g = 18
\][/tex]
2. Rearrange the equation to standard quadratic form:
[tex]\[
g^2 + 3g - 18 = 0
\][/tex]
3. Solve the quadratic equation [tex]\( g^2 + 3g - 18 = 0 \)[/tex]:
To solve the quadratic equation, we can use the quadratic formula, [tex]\( g = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -18 \)[/tex].
Let's calculate the discriminant ([tex]\( \Delta \)[/tex]) first:
[tex]\[
\Delta = b^2 - 4ac = 3^2 - 4(1)(-18) = 9 + 72 = 81
\][/tex]
Now, take the square root of the discriminant:
[tex]\[
\sqrt{\Delta} = \sqrt{81} = 9
\][/tex]
Substitute the values into the quadratic formula:
[tex]\[
g = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-3 \pm 9}{2}
\][/tex]
This gives us two possible solutions:
[tex]\[
g_1 = \frac{-3 + 9}{2} = \frac{6}{2} = 3
\][/tex]
[tex]\[
g_2 = \frac{-3 - 9}{2} = \frac{-12}{2} = -6
\][/tex]
4. Identify the valid solution from the given choices:
The possible value for [tex]\( g \)[/tex] from the multiple-choice options are [tex]\(\{2, 3, 4, 6, 8\}\)[/tex].
Comparing the solutions, [tex]\( g = 3 \)[/tex] is the valid solution within the given choices.
So, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
