Key Properties:
#1: \((x + y)^2 = x^2 + 2xy + y^2\)
#2: \((x - y)^2 = x^2 - 2xy + y^2\)
#3: \(x^2-y^2 = (x+y)(x-y)\)

Given: \(15^2 - 63^2 + 27^2 - 22^2 + (54)(15) + (44)(63)\)

Notice that 15 appears in two terms, and 63 appears in two terms.
Also, if we recognize that 54 = (2)(27), we see that 27 appears twice.
And, if we recognize that 44 = (2)(22), we see that 22 appears twice.

Here's what I mean: \(15^2 - 63^2 + 27^2 - 22^2 + (2)(27)(15) + (2)(63)(22)\)

Let's group the similar terms to get: \(27^2 + (2)(27)(15) +15^2 - 63^2 + (2)(63)(22)-22^2\)
Rewrite to get: \([27^2 + (2)(27)(15) +15^2] - [63^2 - (2)(63)(22)+22^2]\)

Notice that \(27^2 + (2)(27)(15) +15^2\) looks a lot like the expansion in key property #1, and \(63^2 - (2)(63)(22)+22^2\) looks a lot like the expansion in key property #2

So, we can factor them as follows: \((27+15)^2 - (63-22)^2\)
Simplify: \(42^2 – 41^2\) [aha! A difference of squares!!]
Use property #3 to factor the expression: \((42+41)(42-41)\)
Simplify: \((83)(1)\)
Evaluate: \(83\)

Answer: B

Cheers,
Brent
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Hmmm, I figured I'd see a solution by now.

Here are two hints:
1) There is a nice solution that's much faster than simply calculating each part
2) In many cases, the GMAT tests your algebra skills by giving you questions like the above. I wrote an article about it called Common mistake - Not recognizing “hidden” algebra questions : https://www.gmatprepnow.com/articles/co ... -questions

Cheers,
Brent
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Shouldn't it be 83?
(27+15)^2 - (63-22)^2 = 42^2 - 41^2 (42+41)*(42-41) = 83*1 = 83.

Posted from my mobile device

So, adding 42 and 41 was my downfall! Ha!!
Thanks for the heads up!
I've edited my solution.
Kudos for you!!

Cheers,
Brent
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Option B says 63, shouldn't it be 83 ?

Thanks,
K

Is that you, Karaan?
Good catch!! I changed the number in my solution, but not in the original question.
My bad.

Kudos for you!!!
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GMATPrepNow

Yes, it is me

Hmm, that smiley face looks a bit like you, but I thought you had more hair!
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Yeesh, I thought I had more hair . . . until noticed my avatar staring back at me
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