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What is the sum of the solutions of the equation (x-1)^2=|x-1|?

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Math Revolution GMAT Instructor
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What is the sum of the solutions of the equation (x-1)^2=|x-1|?  [#permalink]

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26 Sep 2018, 05:06
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59% (01:40) correct 41% (01:39) wrong based on 198 sessions

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[Math Revolution GMAT math practice question]

What is the sum of the solutions of the equation $$(x-1)^2=|x-1|?$$

$$A. -1$$
$$B. 0$$
$$C. 1$$
$$D. 2$$
$$E. 3$$

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"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Most Helpful Community Reply Director Joined: 13 Mar 2017 Posts: 704 Location: India Concentration: General Management, Entrepreneurship GPA: 3.8 WE: Engineering (Energy and Utilities) What is the sum of the solutions of the equation (x-1)^2=|x-1|? [#permalink] Show Tags Updated on: 04 Oct 2018, 12:06 1 MathRevolution wrote: [Math Revolution GMAT math practice question] What is the sum of the solutions of the equation $$(x-1)^2=|x-1|?$$ $$A. -1$$ $$B. 0$$ $$C. 1$$ $$D. 2$$ $$E. 3$$ $$(x-1)^2=|x-1|$$ For x>=1, $$|x-1| = x-1$$ So, $$(x-1)^2=x-1$$ (x-1)(x-2) = 0 x =1,2 (since both 1,2>=1 . so both solutions are valid) for x<1, $$|x-1| = -(x-1)$$ So, $$(x-1)^2=-(x-1)$$ (x-1)(x) = 0 x =1,0 (since only 0<1. so 0 is a valid solution) So valid solutions for x are 0,1,2 Sum of solutions = 3 $$E. 3$$ _________________ CAT 2017 (98.95) & 2018 (98.91) : 99th percentiler UPSC Aspirants : Get my app UPSC Important News Reader from Play store. MBA Social Network : WebMaggu Appreciate by Clicking +1 Kudos ( Lets be more generous friends.) What I believe is : "Nothing is Impossible, Even Impossible says I'm Possible" : "Stay Hungry, Stay Foolish". Originally posted by shashankism on 03 Oct 2018, 05:01. Last edited by shashankism on 04 Oct 2018, 12:06, edited 1 time in total. General Discussion GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 902 What is the sum of the solutions of the equation (x-1)^2=|x-1|? [#permalink] Show Tags 26 Sep 2018, 05:52 MathRevolution wrote: [Math Revolution GMAT math practice question] What is the sum of the solutions of the equation $$(x-1)^2=|x-1|?$$ $$A. -1$$ $$B. 0$$ $$C. 1$$ $$D. 2$$ $$E. 3$$ $$?\,\,\,:\,\,\,{\text{sum}}\,\,{\text{of}}\,\,{\text{roots}}$$ $${\left( {x - 1} \right)^2} = \left| {x - 1} \right|\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{\left( {x - 1} \right)^4} = {\left( {x - 1} \right)^2}\,\,\,\,\, \Rightarrow \,\,\,\,\left\{ \begin{gathered} \,\,\,x = 1\,\,\,\left( {{\text{trivial}}\,\,{\text{inspection}}} \right) \hfill \\ \,\,{\text{for}}\,\,x \ne 1\,\,:\,\,\,\,\,\frac{{{{\left( {x - 1} \right)}^4}}}{{{{\left( {x - 1} \right)}^2}}} = \frac{{{{\left( {x - 1} \right)}^2}}}{{{{\left( {x - 1} \right)}^2}}}\,\,\,\,\, \Rightarrow \,\,\,\,{\left( {x - 1} \right)^2} = 1\,\,\,\,\, \Rightarrow \,\,\,\,\,x = 0\,\,\,{\text{or}}\,\,\,x = 2 \hfill \\ \end{gathered} \right.$$ (*) When we "square" one equation (or, in general, put it to an EVEN positive power), we don´t loose original roots, but we eventually "add" new ones. That´s why, in the end, we must check each POTENTIAL root in the original equation! All of them fit here. Therefore: $$? = 1 + 0 + 2 = 3$$ This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7086 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: What is the sum of the solutions of the equation (x-1)^2=|x-1|? [#permalink] Show Tags 28 Sep 2018, 00:40 => $$(x-1)^2=|x-1|$$ $$=> |x-1|^2=|x-1|$$ $$=> |x-1|^2-|x-1|=0$$ $$=> |x-1| (|x-1|-1)=0$$ $$=> |x-1| = 0$$ or $$|x-1|-1=0$$ $$=> |x-1| = 0$$ or $$|x-1|=1$$ $$=> x-1 = 0$$ or $$x-1=±1$$ $$=> x=1 = 0$$ or $$x=1±1$$ $$=> x=1, x=0$$ or $$x=2$$ The sum of the solutions is $$0 + 1 + 2 = 3.$$ Therefore, the answer is E. Answer: 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$149 for 3 month Online Course"
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Re: What is the sum of the solutions of the equation (x-1)^2=|x-1|?  [#permalink]

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29 Sep 2018, 07:29
chetan2u Is it wrong if I solve two equations separately and add??

These two:
1) (x-1)^2 = -(x-1)
2) (x-1)^2 = (x-1)

Because I am getting the answer right, but the values I am adding are not the same :/ I'm getting -1 + 1 + 3 =0

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Re: What is the sum of the solutions of the equation (x-1)^2=|x-1|?  [#permalink]

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29 Sep 2018, 09:59
hibobotamuss wrote:
chetan2u Is it wrong if I solve two equations separately and add??

These two:
1) (x-1)^2 = -(x-1)
2) (x-1)^2 = (x-1)

Because I am getting the answer right, but the values I am adding are not the same :/ I'm getting -1 + 1 + 3 =0

-1 and 3 are wrong ..
(X-1)^2=-(x-1).......(x-1)(x-1+1)=0....(x-1)(x)=0, so x=0 or 1
(X-1)^2=x-1......(x-1)(x-1-1)=(x-1)*(x-2)=0...,so x=1 or 2
Different values 0,1 and 2...
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Re: What is the sum of the solutions of the equation (x-1)^2=|x-1|?  [#permalink]

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30 Sep 2018, 11:09
(x-1)^2 = some positive quantity
So, mod(x-1) will be positive or (x-1)will be positive

-----> (x-1)^2= x-1 or x=1 or 2 Sum = 1+2
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What is the sum of the solutions of the equation (x-1)^2=|x-1|?  [#permalink]

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03 Oct 2018, 03:40
2
As for all absolute value questions, we have to consider 2 cases (>0 and <0).

Case 1
$$(x-1)^2=(x-1)$$
(x-1)(x-1)=(x-1) | divide by (x-1)
(x-1)=1

x=2

Case 2
$$(x-1)^2=-(x-1)$$
$$(x-1)^2=-x+1)$$
$$x^2+1-2x=-x+1$$
$$x^2-x=0$$
$$x(x-1)=0$$

x=0, x=1

Sum = 2+1+0=3

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Re: What is the sum of the solutions of the equation (x-1)^2=|x-1|?  [#permalink]

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03 Oct 2018, 08:56
(x-1)^2 = |x-1|
=>x^2-2x+1 = |x-1|
=>x^2-3x+2 Solving it I got x=2 and x=1

Hence E but I am not sure if my method is correct. Can someone guide me please?

Thank you
Arjun
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Re: What is the sum of the solutions of the equation (x-1)^2=|x-1|?  [#permalink]

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03 Oct 2018, 10:46
ArjunJag1328 wrote:
(x-1)^2 = |x-1|
=>x^2-2x+1 = |x-1|
=>x^2-3x+2 Solving it I got x=2 and x=1

Hence E but I am not sure if my method is correct. Can someone guide me please?

Thank you
Arjun

Hi, Arjun!

It´s not correct. Please look at shashankism´s perfect solution (above).
Reason: he started exactly like you. Then he (correctly) considered |x-1| in the two possible scenarios, to finish his solution"shielded".
(You did not find 0, did you realize that?)
Another thing, be careful not to transform an EQUATION into an EXPRESSION (in red). Expressions do not have "roots"!

Regards,
Fabio.
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Re: What is the sum of the solutions of the equation (x-1)^2=|x-1|?   [#permalink] 03 Oct 2018, 10:46
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