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

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Kudos [?]: 4205 [2] , given: 67

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15^2 - 63^2 + 27^2 - 22^2 + (54)(15) + (44)(63) [#permalink] New post 06 Nov 2019, 15:03
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Question Stats:

81% (01:38) correct 18% (03:00) wrong based on 11 sessions
\(15^2 - 63^2 + 27^2 - 22^2 + (54)(15) + (44)(63)=\)

A) 53
B) 83
C) 91
D) 93
E) 105
[Reveal] Spoiler: OA

_________________

Brent Hanneson – Creator of greenlighttestprep.com
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2 KUDOS received
GRE Instructor
User avatar
Joined: 10 Apr 2015
Posts: 3657
Followers: 141

Kudos [?]: 4205 [2] , given: 67

CAT Tests
Re: 15^2 - 63^2 + 27^2 - 22^2 + (54)(15) + (44)(63) [#permalink] New post 24 Mar 2020, 09:01
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GreenlightTestPrep wrote:
\(15^2 - 63^2 + 27^2 - 22^2 + (54)(15) + (44)(63)=\)

A) 53
B) 83
C) 91
D) 93
E) 105


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
_________________

Brent Hanneson – Creator of greenlighttestprep.com
If you enjoy my solutions, you'll love my GRE prep course!
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Re: 15^2 - 63^2 + 27^2 - 22^2 + (54)(15) + (44)(63)   [#permalink] 24 Mar 2020, 09:01
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