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Sagot :
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
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
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