[email protected] wrote:What is the sum of all possible solutions of the equation |x + 4|² - 10|x + 4| = 24?
-16
-14
-12
-8
-6
|x + 4|² = (x+4)² = x² + 8x + 16.
Replacing |x + 4|² with x² + 8x + 16, we get:
x² + 8x + 16 - 10|x+4| = 24
x² + 8x - 8 - 10|x+4| = 0.
|x+4| must represent a NONNEGATIVE value.
Case 1: x≥-4
In this case, |x+4| = x+4.
To illustrate:
If x=-3, then |x+4| = x+4 = -3+4 = 1.
Case 2: x<-4
In this case, |x+4| = -(x+4) = -x-4.
To illustrate:
If x=-5, then |x+4| = -x-4 = -(-5) - 4 = 1.
Case 1: x≥-4
Replacing |x+4| with x+4, we get:
x² + 8x - 8 - 10(x+4) = 0
x² + 8x - 8 - 10x - 40 = 0
x² - 2x - 48 = 0
(x-8)(x+6) = 0
x=8 or x=-6.
Since the tested range is x≥-4, only x=8 is within the required range.
Thus, x=8.
Case 2: x<-4
Replacing |x+4| with -x-4, we get:
x² + 8x - 8 - 10(-x-4) = 0
x² + 8x - 8 + 10x + 40 = 0
x² + 18 + 32 = 0
(x+16)(x+2) = 0
x=-16 or x=-2.
Since the tested range is x<-4, only x=-16 is within the required range.
Thus, x=-16.
Sum of the solutions = 8 + (-16) = -8.
The correct answer is
D.
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