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Intern  Joined: 30 Dec 2016
Posts: 1

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1
I think this is a high-quality question and I agree with explanation. real good question
Intern  B
Joined: 31 Jul 2013
Posts: 15
Location: Viet Nam
Concentration: General Management, Entrepreneurship
GMAT 1: 650 Q49 V28 GPA: 3.46
WE: Sales (Computer Software)

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I think this is a high-quality question and I agree with explanation.
Intern  B
Joined: 03 May 2014
Posts: 16

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Harley1980 wrote:
mejia401 wrote:
The clue helped, but I still couldn't bridge how to eliminate the other variables with what I had. The logic which I'm to substitute or combine is kind of ambiguous now, and this problem exploited that weakness. I found the article below, but any body have other guidance on how to step through expressions in a logical manner?

http://gmatclub.com/blog/2010/11/should-i-use-combination-or-substitution/

Thanks,

Hello mejia401

This is quite painful topic for me too. And I still struggle with it occasionally but I'll try to help to you with this one:

firstly we have three equations:
$$P^2 - QR = 10$$
$$Q^2 + PR = 10$$
$$R^2 + PQ = 10$$

We see that each pair of them have three pair of the same values. Let's take for example another pair (not which Bunuel describe in his solution)
The method will be same:

$$P^2 - QR = 10$$
$$Q^2 + PR = 10$$

our goal is to get rid of squares and received clear values
we can make it by using formula $$a^2-b^2=(a-b)(a+b)$$

here is a lot about algebra and exactly about these Algebraic Identities
http://gmatclub.com/forum/math-algebra-101576.html (at the end of the article in subsection Algebraic Identities)

So we need to make from these two equations
$$P^2 - QR = 10$$
$$Q^2 + PR = 10$$

one equation without exponents
and we can see that by subtraction we will receive $$Q^2-P^2$$ so we can try to check this idea:
$$Q^2 + PR - P^2 + QR = 0$$
$$Q^2 - P^2 + PR + QR = 0$$
$$(Q - P)(Q + P) + PR + QR = 0$$

Now we need to receive $$Q - P$$ or $$Q + P$$ for cancellation this equation
from this part $$PR + QR$$ we can receive $$R(Q+P)$$
So we have:
$$(Q - P)(Q + P) - R(Q+P) = 0$$ --> $$(Q - P)(Q + P) = -R(Q+P)$$ --> cancel $$(Q + P)$$ from both sides --> $$(Q - P) = -R$$ or $$P = R + Q$$

And now we should see the way to eliminate these obnoxious parts $$PR$$ and $$PQ$$
Again we should take a look on all three equations and try to see what we can eliminate to receive needed result.
$$P^2 - QR = 10$$
$$Q^2 + PR = 10$$
$$R^2 + PQ = 10$$

We have part $$Q+R = P$$ so ideally will be to find parts in which we have the same disguised expression
These two equations have it
$$Q^2 + PR = 10$$
$$R^2 + PQ = 10$$

Let's sum them: $$Q^2 + R^2 + PR + PQ = 20$$ --> undisguise $$P + Q$$ --> $$Q^2 + R^2 + P(R + Q) = 20$$
And now we have expression from our question $$Q^2 + R^2 + P^2 = 20$$

____

For solving such tasks you should know Algebraic identites by heart and do a lot of such tasks with thinking about why author of solution pick exactly these values to cancel or to transform in another view

The best way of practice is to start solving all tasks from section Algebra.
http://gmatclub.com/forum/gmat-ps-quest ... 27957.html

I think after 50-70 solved and dissected tasks you will obtain "feeling" of "where is hide identities which will help to solve the task" and this process even can became boring for 600 lvl tasks When you move the highlighted expression over to the other side the expression moves from a negative to a positive. Therefore $$Q - P = R$$ Which would change your solution.
Manager  G
Joined: 10 Mar 2016
Posts: 149
Location: India
Concentration: Finance, Real Estate
GMAT 1: 550 Q44 V23 WE: Corporate Finance (Commercial Banking)

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I think this is a high-quality question and I agree with explanation. Excellent question.
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Intern  B
Joined: 20 Aug 2016
Posts: 49
GMAT 1: 570 Q46 V23 GMAT 2: 610 Q49 V25 GMAT 3: 620 Q45 V31 WE: Information Technology (Other)

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Nawfal, read the ques again, your equation is incorrect and hence the solution too.
Intern  G
Joined: 21 May 2016
Posts: 28

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$$P^2-QR=10----1$$
$$R^2+PQ=10----2$$
Subtract 2 from 1
$$P^2-R^2-QR-PQ=0$$
$$P^2-R^2=Q(R+P)$$
(P-R)(P+R)=Q(R+P)
P-R=Q
===>P-Q=R
Also,
$$P^2+Q^2+PR-QR=20$$
$$P^2+Q^2+R(P-Q)=20$$
$$P^2+Q^2+R^2=20$$
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Intern  B
Joined: 09 Nov 2017
Posts: 3

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I think this is a high-quality question and I agree with explanation.
Senior Manager  G
Joined: 22 Nov 2018
Posts: 485
Location: India
GMAT 1: 640 Q45 V35 GMAT 2: 660 Q48 V33 ### Show Tags

I think this is a high-quality question and I agree with explanation.
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Intern  B
Joined: 24 Dec 2018
Posts: 36

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I think this question can be wrapped up quite easily within a minute if you use the hint appropriately. It is given that Q is not equal to R but nowhere given that P is not equal to R and the beauty of equations with variables is that they should satisfy every condition not specified explicitly. So just replace R with P and voila!!
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Intern  B
Joined: 30 May 2017
Posts: 32
Concentration: Marketing, General Management
GPA: 3.96
WE: Engineering (Health Care)

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After struggling for some time with this, I finally marked it on the basis that what can be the sum of 3 numbers.

20 = 16+4+0

Marked it correct  Intern  B
Joined: 15 Jul 2016
Posts: 27
GMAT 1: 690 Q48 V36 ### Show Tags

I think this is a high-quality question and I agree with explanation.

Hi, can somebody please explain me what is the significance of R≠Q here?

I solved this question without paying much heed to this information, and looking at this forum here, it seems that this part of the question was very valuable in solving it. What hint does it provide?

Help will be appreciated!!

Thanks _________________
Please give Kudos if you agree with my approach :) Re M06-08   [#permalink] 03 Jul 2019, 00:57

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# M06-08

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