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What is the product of all roots of the equation

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What is the product of all roots of the equation

by Max@Math Revolution » Tue Jan 16, 2018 1:25 am
[GMAT math practice question]

What is the product of all roots of the equation (x+1)^2 = l x+1l ?

A. -2
B. -1
C. 0
D. 1
E. 2
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by GMATGuruNY » Tue Jan 16, 2018 4:42 am
Max@Math Revolution wrote:[GMAT math practice question]

What is the product of all roots of the equation (x+1)^2 = l x+1l ?

A. -2
B. -1
C. 0
D. 1
E. 2
A quick test of easy integer values (0, 1, -1, etc.) reveals that x=0 is one of the roots for (x+1)² = |x+1|:
(0+1)² = |0+1|
1 = 1.

Thus, the other roots are irrelevant.
Since one of the roots is 0, the product of the roots must also be 0.

The correct answer is C.
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by Brent@GMATPrepNow » Tue Jan 16, 2018 7:45 am
Max@Math Revolution wrote: What is the product of all roots of the equation (x + 1)² = | x + 1| ?

A. -2
B. -1
C. 0
D. 1
E. 2
Another approach:

There are 3 steps to solving equations involving ABSOLUTE VALUE:
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

Given: (x + 1)² = | x + 1|
Apply rule to get two equations: (x + 1)² = x + 1 and -(x + 1)² = x + 1

Take: (x + 1)² = x + 1
Expand and simplify left side: x² + 2x + 1 = x + 1
Set this quadratic equation to equal zero: x² + x = 0
Factor to get: x(x + 1) = 0
So, x = 0 and x = -1 are two solutions (aka roots) of the original equation
When we test these two solutions, we find that they BOTH work.

IMPORTANT: At this point, we COULD solve -(x + 1)² = x + 1 for x also. HOWEVER, doing so would be a waste of time since the questions asks us to find the PRODUCT of all possible solutions.
Since x = 0 is one of the solutions, we can be sure that the product will be 0

Answer: C

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EDIT

by Max@Math Revolution » Thu Jan 18, 2018 12:33 am
=>
Now,
(x+1)^2=|x+1|
⇔ |x+1|^2=|x+1|
⇔ |x+1|^2-|x+1|= 0
⇔ |x+1|(|x+1|-1) = 0
⇔ |x+1| = 0 or |x+1| = 1
⇔ x = -1 or x+1 = ±1
⇔ x = -1 or x = -1 ±1
⇔ x = -1, x= -2 or x = 0

The product of these solutions is (-1)*(-2)*0 = 0.
Therefore, the answer is C.
Answer: C
Last edited by Max@Math Revolution on Tue Jan 23, 2018 10:23 pm, edited 1 time in total.
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by Scott@TargetTestPrep » Thu Jan 18, 2018 1:34 pm
Max@Math Revolution wrote:[GMAT math practice question]

What is the product of all roots of the equation (x+1)^2 = l x+1l ?

A. -2
B. -1
C. 0
D. 1
E. 2
Let's simplify the given equation:

x^2 + 2x + 1 = lx+1l

First we can solve for when x+1 is positive:

x^2 + 2x + 1 = x + 1

x^2 + x = 0

x(x + 1) = 0

x = 0 or x = -1

Since we already see that x = 0 (i.e., one of the roots is 0), we know that the product of all the roots will always be 0 also. We don't need to finish the complete solution to the problem.

Answer: C

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