To determine which of the potential roots are actual roots of the function [tex]\( f(x) = 2x^2 + 2x - 24 \)[/tex], we need to check each one individually by substituting them into the function and determining if the function value is zero at that point.
Let's evaluate [tex]\( f(x) \)[/tex] for each given potential root:
1. For [tex]\( x = -4 \)[/tex]:
[tex]\[
f(-4) = 2(-4)^2 + 2(-4) - 24
\][/tex]
[tex]\[
= 2(16) + 2(-4) - 24
\][/tex]
[tex]\[
= 32 - 8 - 24
\][/tex]
[tex]\[
= 0
\][/tex]
Thus, [tex]\( x = -4 \)[/tex] is an actual root.
2. For [tex]\( x = -3 \)[/tex]:
[tex]\[
f(-3) = 2(-3)^2 + 2(-3) - 24
\][/tex]
[tex]\[
= 2(9) + 2(-3) - 24
\][/tex]
[tex]\[
= 18 - 6 - 24
\][/tex]
[tex]\[
= -12
\][/tex]
Thus, [tex]\( x = -3 \)[/tex] is not an actual root.
3. For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2(2)^2 + 2(2) - 24
\][/tex]
[tex]\[
= 2(4) + 2(2) - 24
\][/tex]
[tex]\[
= 8 + 4 - 24
\][/tex]
[tex]\[
= -12
\][/tex]
Thus, [tex]\( x = 2 \)[/tex] is not an actual root.
4. For [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 2(3)^2 + 2(3) - 24
\][/tex]
[tex]\[
= 2(9) + 2(3) - 24
\][/tex]
[tex]\[
= 18 + 6 - 24
\][/tex]
[tex]\[
= 0
\][/tex]
Thus, [tex]\( x = 3 \)[/tex] is an actual root.
5. For [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 2(4)^2 + 2(4) - 24
\][/tex]
[tex]\[
= 2(16) + 2(4) - 24
\][/tex]
[tex]\[
= 32 + 8 - 24
\][/tex]
[tex]\[
= 16
\][/tex]
Thus, [tex]\( x = 4 \)[/tex] is not an actual root.
From our evaluations, we conclude that the actual roots of the function [tex]\( f(x) = 2x^2 + 2x - 24 \)[/tex] are [tex]\( x = -4 \)[/tex] and [tex]\( x = 3 \)[/tex].
Therefore, the correct answer is:
-4 and 3
