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# The figure shows a square patio surrounded by a walkway of

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Director
Joined: 03 Nov 2004
Posts: 868
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Kudos [?]: 19 [0], given: 0

The figure shows a square patio surrounded by a walkway of [#permalink]  30 Jun 2005, 19:03
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The figure shows a square patio surrounded by a walkway of width x meters. If the area of the walkway is 132 square meters and the width of the patio is 5 meters greater than the width of the walkway, what is the area of the patio, in square meters?
A. 56
B. 64
C. 68
D. 81
E. 100
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Senior Manager
Joined: 30 May 2005
Posts: 374
Followers: 1

Kudos [?]: 7 [0], given: 0

Clearly the patio is a square, since all of its sides of off 2x from the bigger square.

Let the patio's side be a. => x=a-5

Side of bigger square = a + 2(a-5) = 3a - 10

The area of the walkway = Area of big square - Area of patio

132 = (3a-10)^2 - a^2

132 = 9a^2 -60a + 100 -a^2

This reduces to:

8a^2 - 60a - 32 = 0
2a^2 -15a - 8 = 0
2a^2 + a -16a - 8 = 0
a(2a + 1) -8(2a+1) = 0
(a-8)*(2a+1) = 0
a = 8 or -1/2, But a is positive => a=8

Patio's area = a^2 = 64

Director
Joined: 03 Nov 2004
Posts: 868
Followers: 1

Kudos [?]: 19 [0], given: 0

AJB77 wrote:
2a^2 -15a - 8 = 0
2a^2 + a -16a - 8 = 0
a(2a + 1) -8(2a+1) = 0
(a-8)*(2a+1) = 0
a = 8 or -1/2, But a is positive => a=8

Patio's area = a^2 = 64

Can you tell me how you deduced that. Is it 2 * -8 = -16
Senior Manager
Joined: 30 May 2005
Posts: 374
Followers: 1

Kudos [?]: 7 [0], given: 0

Can you tell me how you deduced that. Is it 2 * -8 = -16

Yes. After a few practice factorizations, it becomes easier to spot.
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