What is the Product of all Possible Solutions of the Equation \(|x+2|^2\)- 5|x+2| = -6? GMAT Problem Solving

Question: What is the product of all possible solutions of the equation |x+2|\(^2\)- 5|x+2| = -6?

1. -20
2. 5
3. -4
4. 0
5. 20

“What is the product of all possible solutions of the equation |x+2|\(^2\) - 5|x+2| = -6? “- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”.
To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical understanding.

Answer

Approach Solution 1

Denote |x+2| as a

We get \(a^2-5a=-6\)

Solving quadratics give a = 3 or a = 2

Thus

|x+2| = 3 or |x+2| = 2

Further solving gives, x = 1, x = -5, x = 0, or x = -4

The product of all those values is 0

Correct option: D

Approach Solution 2

|x+2|\(^2\) - 5|x+2| = -6

Let us assume |x+2| = y

So, our equation will become:

\(y^2-5y+6=0\)

Factorize the equation and we will get:

(y - 3) (y - 2) = 0

So, value of y = 3, 2

As y = |x+2|

So putting values of y will give the following results:

When y = 3

|x+2| = 3

x+2 = +3 or -3

So, x = 1 or -5

When y = 2

|x+2| = 2

x+2 = +2 or -2

x = 0 or -4

So product of all the possible solutions of the given equation is = (1) * (-5) * (0) * (-4) = 0

Correct option: D

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