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What is the sum of all solutions to the equation

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What is the sum of all solutions to the equation [#permalink] New post 13 Nov 2016, 09:06
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Question Stats:

30% (01:41) correct 70% (01:50) wrong based on 10 sessions
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
[Reveal] Spoiler: OA

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Re: What is the sum of all solutions to the equation [#permalink] New post 13 Nov 2016, 09:07
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Expert's post
GreenlightTestPrep 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:
[Reveal] Spoiler:
D


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Re: What is the sum of all solutions to the equation [#permalink] New post 18 Dec 2016, 08:35
This is kind of a bad question because people who neglect the absolute value rule will accidentally get the right answer.
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Re: What is the sum of all solutions to the equation [#permalink] New post 24 Jul 2017, 06:59
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Is a really bad question. The time is to short to check for extraneous roots. I did the exercise well, but I totally forget to check if the values were correct. Bad for me.
Re: What is the sum of all solutions to the equation   [#permalink] 24 Jul 2017, 06:59
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