Problem Solving

What is the sum of all solutions to the equation |x² - 4x + 4| = x² + 10x - 24?

A) -5
B) -3
C) -2
D) 2
E) 5

Here's a 600-level question to try.

Answer: D

Brent@GMATPrepNow wrote:What is the sum of all solutions to the equation |x² - 4x + 4| = x² + 10x - 24?

A) -5
B) -3
C) -2
D) 2
E) 5

When solving equations involving ABSOLUTE VALUE, there are 3 steps:
1. Apply the rule that says: If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug solutions into original equation to check for extraneous roots

So, we have two equations to solve: x² - 4x + 4 = x² + 10x - 24 and x² - 4x + 4 = -(x² + 10x - 24)

x² - 4x + 4 = x² + 10x - 24
Subtract x² from both sides: -4x + 4 = 10x - 24
Rearrange: 28 = 14x
Solve: x = 2

x² - 4x + 4 = -(x² + 10x - 24)
Simplify right side: x² - 4x + 4 = -x² - 10x + 24
Add x² to both sides: 2x² - 4x + 4 = -10x + 24
Add 10x to both sides: 2x² + 6x + 4 = 24
Subtract 24 from both sides: 2x² + 6x - 20 = 0
Factor: 2(x² + 3x - 10) = 0
Factor again: 2(x - 2)(x + 5) = 0
Solve: x = 2 and x = -5

So, we have two solutions to consider: x = 2 and x = -5
Plug solutions into original equation to check for extraneous roots

x = 2
|2² - 4(2) + 4| = 2² + 10(2) - 24
Evaluate: |0| = 0
This works, so keep this solution

x = -5
|(-5)² - 4(-5) + 4| = (-5)² + 10(-5) - 24
Evaluate: |49| = -49
Doesn't work. So, x = -5 is NOT a solution

Since there's only one valid solution (x = 2), the sum of all solutions is 2.
Answer: D

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