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

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GRE Instructor
Joined: 10 Apr 2015
Posts: 3006
Followers: 112

Kudos [?]: 3303 [1] , given: 56

15^2 - 63^2 + 27^2 - 22^2 + (54)(15) + (44)(63) [#permalink]  06 Nov 2019, 15:03
1
KUDOS
Expert's post
00:00

Question Stats:

85% (00:54) correct 14% (01:05) wrong based on 7 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

GRE Instructor
Joined: 10 Apr 2015
Posts: 3006
Followers: 112

Kudos [?]: 3303 [1] , given: 56

Re: 15^2 - 63^2 + 27^2 - 22^2 + (54)(15) + (44)(63) [#permalink]  24 Mar 2020, 09:01
1
KUDOS
Expert's post
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: $$83$$

Cheers,
Brent
_________________

Brent Hanneson – Creator of greenlighttestprep.com

Re: 15^2 - 63^2 + 27^2 - 22^2 + (54)(15) + (44)(63)   [#permalink] 24 Mar 2020, 09:01
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