[email protected] wrote:What is the sum of all roots of the equation
|x + 4|^2 - 10|x + 4| = 24?
Alternate approach:
|x+4| represents a NONNEGATIVE value.
Case 1: x≥-4
Here, |x+4| = x+4.
To illustrate:
If x=-3, then |x+4| = x+4 = -3+4 = 1.
The result is a nonnegative value.
Case 2: x<-4
Here, |x+4| = -x-4.
To illustrate:
If x=-5, then |x+4| = -x-4 = -(-5)-4 = 1.
The result is a nonnegative value.
Solve for each case.
Case 1: x≥-4, implying that |x+4| = x+4
(x+4)² - 10(x+4) = 24
x² + 8x + 16 - 10x - 40 - 24 = 0
x² - 2x - 48 = 0
(x+6)(x-8) = 0.
x=-6 or x=8.
But x=-6 is not valid, since the range here is x≥-4.
Thus, the only valid solution in Case 1 is x=8.
Case 2: x<-4, implying that |x+4| = -x-4
(-x-4)² - 10(-x-4) = 24
x² + 8x + 16 + 10x + 40 - 24 = 0
x² + 18x + 32 = 0
(x+2)(x+16) = 0
x=-2 or x=-16.
But x=-2 is not valid, since the range here is x<-4.
Thus, the only valid solution in Case 2 is x=-16.
Sum of the solutions = 8 + (-16) = -8.
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