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
RELATED VIDEO
- Solving equations with absolute value:
https://www.gmatprepnow.com/module/gmat ... /video/972
