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What is the product of all roots of the equation (x

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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GPA: 3.82
What is the product of all roots of the equation (x [#permalink]

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16 Jan 2018, 01:27
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[GMAT math practice question]

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

$$A. -2$$
$$B. -1$$
$$C. 0$$
$$D. 1$$
$$E. 2$$
[Reveal] Spoiler: OA

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Last edited by Bunuel on 16 Jan 2018, 10:09, edited 1 time in total.
Edited the OA.
Math Expert
Joined: 02 Aug 2009
Posts: 5658
Re: What is the product of all roots of the equation (x [#permalink]

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16 Jan 2018, 06:02
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Expert's post
MathRevolution wrote:
[GMAT math practice question]

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

$$A. -2$$
$$B. -1$$
$$C. 0$$
$$D. 1$$
$$E. 2$$

Hi...

you are required to relook into the OA..
$$(x+1)^2=|x+1|$$ at the first look will give 0 as a root..
x as 0 will make the equation... $$(x+1)^2=|x+1|........(0+1)^2=|0+1|....1=1$$
so product of all roots irrespective of other roots will remain ZERO
C..

if you want to solve it..

when x+1 is negative..
$$(x+1)^2=|x+1|............x^2+2x+1=-x-1......x^2+3x+2=0.......(x+2)(x+1)=0$$
roots are -2 and -1

when x+1 is positive..
$$(x+1)^2=|x+1|............x^2+2x+1=x+1......x^2+x=0.......x(x+1)=0$$
roots are 0 and -1...

product : 0*-1*-2=0
C
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Re: What is the product of all roots of the equation (x [#permalink]

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16 Jan 2018, 07:46
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MathRevolution wrote:
[GMAT math practice question]

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

$$A. -2$$
$$B. -1$$
$$C. 0$$
$$D. 1$$
$$E. 2$$

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

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Re: What is the product of all roots of the equation (x [#permalink]

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16 Jan 2018, 10:05
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I think OA should be C instead of D.
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Joined: 02 Sep 2009
Posts: 43850
Re: What is the product of all roots of the equation (x [#permalink]

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16 Jan 2018, 10:10
sghoshgt wrote:
I think OA should be C instead of D.

Yes, you are right, it's C. Edited. Thank you.
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Re: What is the product of all roots of the equation (x [#permalink]

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16 Jan 2018, 10:29
Consider x+1 = A

so A^2 = |A|

Squaring both the sides

A^4 = A^2 ( this will always be positive )

A^4 - A^2 = 0

A^2 ( A^2 - 1 ) = 0

A=0,1,-1

Product of roots = 0
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Joined: 26 Mar 2013
Posts: 1430
Re: What is the product of all roots of the equation (x [#permalink]

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17 Jan 2018, 03:03
saraheja wrote:
Consider x+1 = A

so A^2 = |A|

Squaring both the sides

A^4 = A^2 ( this will always be positive )

A^4 - A^2 = 0

A^2 ( A^2 - 1 ) = 0

A=0,1,-1

Product of roots = 0

While The answer is correct, you must return to original equation you have (x+1=A) to find values of x. It is not not about A.
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 4896
GPA: 3.82
What is the product of all roots of the equation (x [#permalink]

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18 Jan 2018, 00:42
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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 $$(-1)*(-2)*0 = 0$$.

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What is the product of all roots of the equation (x   [#permalink] 18 Jan 2018, 00:42
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