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# What is the product of all the roots on the equation (x-2)^2=|x-2|?

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Math Revolution GMAT Instructor
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What is the product of all the roots on the equation (x-2)^2=|x-2|? [#permalink]

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28 Sep 2017, 02:34
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[GMAT math practice question]

What is the product of all the roots on the equation (x-2)^2=|x-2|?

A. 2
B. -2
C. 3
D. -3
E. 6
[Reveal] Spoiler: OA

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"Only $79 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" PS Forum Moderator Joined: 25 Feb 2013 Posts: 1012 Location: India GPA: 3.82 What is the product of all the roots on the equation (x-2)^2=|x-2|? [#permalink] ### Show Tags 28 Sep 2017, 02:52 MathRevolution wrote: [GMAT math practice question] What is the product of all the roots on the equation (x-2)^2=|x-2|? A. 2 B. -2 C. 3 D. -3 E. 6 Case 1: If $$x-2>0$$, then $$(x-2)^2=(x-2)$$ or $$(x-2)^2-(x-2)=0 =>(x-2)(x-2-1)=0$$ Hence the roots are $$x-2=0$$, or $$x=2$$ and $$x-3=0$$, or $$x=3$$ Case 2 If $$x-2<0$$, then $$(x-2)^2=-(x-2)$$ or $$(x-2)^2+(x-2)=0 =>(x-2)(x-2+1)=0$$ Hence the roots are $$x-2=0$$, or $$x=2$$ and $$x-1=0$$, or $$x=1$$ So, combining Case 1 & Case 2, the roots of the equation are $$1$$, $$2$$ and $$3$$ Product of roots $$=1*2*3=6$$ Option E ------------------------------------------------------- Another method $$(x-2)^2=|x-2|$$, square both sides to get $$(x-2)^4=(x-2)^2$$ or $$(x-2)^4-(x-2)^2=0 =>(x-2)^2(x-3)(x-1)=0$$ So roots of the equation are $$(x-2)^2=0$$, $$x-3=0$$ and $$x-1=0$$ Hence $$x = 1$$, $$2$$ and $$3$$ So Product of roots $$= 1*2*3$$ Last edited by niks18 on 30 Sep 2017, 02:36, edited 1 time in total. Intern Joined: 30 Aug 2017 Posts: 4 Re: What is the product of all the roots on the equation (x-2)^2=|x-2|? [#permalink] ### Show Tags 28 Sep 2017, 08:01 Hi MathRevolution, I could work out one of the integer the x = 2, however couldn't work out how to get the -3. Where does the the -1 come from in the operation (x-2-1)? (x−2)2−(x−2)=0=>(x−2)(x−2−1)=0 Thank you, Nick PS Forum Moderator Joined: 25 Feb 2013 Posts: 1012 Location: India GPA: 3.82 Re: What is the product of all the roots on the equation (x-2)^2=|x-2|? [#permalink] ### Show Tags 28 Sep 2017, 08:05 nmargot wrote: Hi MathRevolution, I could work out one of the integer the x = 2, however couldn't work out how to get the -3. Where does the the -1 come from in the operation (x-2-1)? (x−2)2−(x−2)=0=>(x−2)(x−2−1)=0 Thank you, Nick Hi nmargot in the expression $$(x-2)^2-(x-2)$$, I have taken $$(x-2)$$ as common So $$(x-2)^2-1*(x-2) = (x-2)(x-2-1)$$ Hope this is clear Intern Joined: 23 Jan 2017 Posts: 3 Re: What is the product of all the roots on the equation (x-2)^2=|x-2|? [#permalink] ### Show Tags 29 Sep 2017, 07:04 for x<2 the equation becomes: $$(x-2)^2 = (2-x)$$ on solving: x=1 or x=2(not possible, since x<2) for x>2 the equation can be rewritten as: $$(x-2)^2 = (x-2)$$ on solving: x=2(not possible, since x>2) or x=3 for x=2 $$(2-2)^2 = (2-2)$$; which satisfies the equation. Hence the roots are 1,2,3 product = 6 PS Forum Moderator Joined: 25 Feb 2013 Posts: 1012 Location: India GPA: 3.82 Re: What is the product of all the roots on the equation (x-2)^2=|x-2|? [#permalink] ### Show Tags 29 Sep 2017, 21:44 1 This post received KUDOS naman90 wrote: for x<2 the equation becomes: $$(x-2)^2 = (2-x)$$ on solving: x=1 or x=2(not possible, since x<2) for x>2 the equation can be rewritten as: $$(x-2)^2 = (x-2)$$ on solving: x=2(not possible, since x>2) or x=3 for x=2 $$(2-2)^2 = (2-2)$$; which satisfies the equation. Hence the roots are 1,2,3 product = 6 Hi naman90 and MathRevolution I have two fundamental disconnect here with the solution, Kindly explain 1.) Any 2nd degree polynomial will have only two roots, but here we are getting $$3$$ roots. Is this possible? 2.) the equation is of the form $$a^2=|a|$$, or $$a^2=±a$$. Now we know that $$a^2$$ or LHS is always positive, so RHS cannot be negative, hence $$a^2≠-a$$. Hence the only solution is $$a^2=a$$. But as per the solution provided $$a^2=-a$$ is yielding as possible root $$1$$. what am I missing here? Intern Joined: 23 Jan 2017 Posts: 3 What is the product of all the roots on the equation (x-2)^2=|x-2|? [#permalink] ### Show Tags 30 Sep 2017, 00:17 hi niks18 although i am no math expert but here is my take on the points you raised 1. introducing modulus to an equation increases the number of possible solutions. take |x| = 2 for example. even though it is a 1st degree equation, it has 2 roots (2 and -2) 2. when you say "Now we know that $$a^2$$ or LHS is always positive, so RHS cannot be negative, hence $$a^2≠−a$$", you are making an assumption that a>0. you also have to solve the equation for the case when a<0. PS Forum Moderator Joined: 25 Feb 2013 Posts: 1012 Location: India GPA: 3.82 Re: What is the product of all the roots on the equation (x-2)^2=|x-2|? [#permalink] ### Show Tags 30 Sep 2017, 02:36 naman90 wrote: hi niks18 although i am no math expert but here is my take on the points you raised 1. introducing modulus to an equation increases the number of possible solutions. take |x| = 2 for example. even though it is a 1st degree equation, it has 2 roots (2 and -2) 2. when you say "Now we know that $$a^2$$ or LHS is always positive, so RHS cannot be negative, hence $$a^2≠−a$$", you are making an assumption that a>0. you also have to solve the equation for the case when a<0. Thanks naman90 for highlighting a key fact that i completely missed. Actually the equation is a 4th degree polynomial and hence will have 4 roots = 1,2 (twice) & 3 because to remove mod we square it, hence the equation becomes a 2nd degree polynomial, hence two roots. For ex $$|x|=2$$, $$x^2=4$$ Thanks again, was lucky to get the answer, but will have to edit the solution. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5079 GMAT 1: 800 Q59 V59 GPA: 3.82 What is the product of all the roots on the equation (x-2)^2=|x-2|? [#permalink] ### Show Tags 01 Oct 2017, 18:17 => (x-2)^2=|x-2| ⬄ |x-2|^2=|x-2| ⬄ |x-2|^2-|x-2| = 0 ⬄ |x-2| ( |x-2| - 1 ) = 0 ⬄ |x-2| = 0 or |x-2| = 1 ⬄ x = 2 or x-2 = ±1 ⬄ x = 2 or x = 2 ±1 ⬄ x = 2 or x = 3 or x = 1 1 * 2 * 3 = 6 Ans: E _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 3 month Online Course"
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What is the product of all the roots on the equation (x-2)^2=|x-2|?   [#permalink] 01 Oct 2017, 18:17
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