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# In the above figure, PQRS is a square with side x + 4. Each of the

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Math Expert
Joined: 02 Sep 2009
Posts: 43813
In the above figure, PQRS is a square with side x + 4. Each of the [#permalink]

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20 Sep 2017, 00:53
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In the above figure, PQRS is a square with side x + 4. Each of the four smaller squares has side of length 2. If the area of the shaded region is 48, what is the value of x?

(A) 1
(B) 4
(C) 4√2
(D) 8
(E) 12

[Reveal] Spoiler:
Attachment:

2017-09-20_1020.png [ 4.39 KiB | Viewed 584 times ]
[Reveal] Spoiler: OA

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Senior Manager
Status: Preparing for GMAT
Joined: 25 Nov 2015
Posts: 481
Location: India
GPA: 3.64
Re: In the above figure, PQRS is a square with side x + 4. Each of the [#permalink]

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20 Sep 2017, 04:52
Bunuel wrote:

In the above figure, PQRS is a square with side x + 4. Each of the four smaller squares has side of length 2. If the area of the shaded region is 48, what is the value of x?

(A) 1
(B) 4
(C) 4√2
(D) 8
(E) 12

[Reveal] Spoiler:
Attachment:
2017-09-20_1020.png

Width of the shaded portion = x-4-2-2=x (subtracting 2m from each side)
Length = 2
Area of each shaded rectangle = 2x
Area of 4 shaded rectangles = 4 x2x = 8x
Area of the middle square portion of shade = x^2
Total shaded area = x^2+8x = 48
Solving the quadratic equation, x =4

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VP
Joined: 22 May 2016
Posts: 1330
In the above figure, PQRS is a square with side x + 4. Each of the [#permalink]

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20 Sep 2017, 05:13
Bunuel wrote:

In the above figure, PQRS is a square with side x + 4. Each of the four smaller squares has side of length 2. If the area of the shaded region is 48, what is the value of x?

(A) 1
(B) 4
(C) 4√2
(D) 8
(E) 12

[Reveal] Spoiler:
Attachment:
2017-09-20_1020.png

Find numeric value for area of large square. Use (x + 4) to derive equation for the same area. Set them equal to solve for $$x$$.

Numeric area

(Area of small squares) + (Area of shaded region) = Area of large square

Small square side = 2. Each small square's area = 4. Their total area = 4*4 = 16

Area of shaded region = 48

Area of large square = 16 + 48 = 64

Equation for same area

One side of large square = $$(x + 4)$$
Equation for area of large square is
$$(x + 4)^{2}$$ = $$x^2 + 8x + 16$$

From above, numeric area of large square = 64. Set equation equal to numeric value:

$$x^2 + 8x + 16 = 64$$
$$x^2 + 8x - 48 = 0$$
$$(x + 12)(x - 4) = 8$$
$$x$$ cannot be negative
$$x = 4$$

Edited to correct a really dumb mistake! Answer changed.
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Formerly genxer123

Last edited by generis on 20 Sep 2017, 10:57, edited 1 time in total.
Senior Manager
Status: Preparing for GMAT
Joined: 25 Nov 2015
Posts: 481
Location: India
GPA: 3.64
Re: In the above figure, PQRS is a square with side x + 4. Each of the [#permalink]

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20 Sep 2017, 09:54
1
KUDOS
genxer123 wrote:
Bunuel wrote:

In the above figure, PQRS is a square with side x + 4. Each of the four smaller squares has side of length 2. If the area of the shaded region is 48, what is the value of x?

(A) 1
(B) 4
(C) 4√2
(D) 8
(E) 12

[Reveal] Spoiler:
Attachment:
2017-09-20_1020.png

Find the area of the large square in variable form. Subtract the total area of the four small squares. That equals the area of shaded region.

One side of large square = (x + 4)

One side of small square = 2

Each large square side includes two sides of small square, so large square side =
x + 4 - 2 - 2 = x

Area of large square: $$x^2$$

Area of one small square (2*2) = 4

Total area of four small squares = 16

Area of shaded region, given = 48

$$(Area_{large}) - (Area_{4Small}) = (Area_{shaded})$$

$$x^2 - 16 = 48$$
$$x^2 = 64$$
$$x = 8$$

Hi Genxer123
Please check the red font portion marked above.
Area of the large square shall be $$(x+4)^2$$
and x is the length of the side of rectangular portion of shaded area.
Please refer my solution in the previous post.

Please give kudos, if u liked my post!
_________________

Please give kudos, if you like my post

When the going gets tough, the tough gets going...

VP
Joined: 22 May 2016
Posts: 1330
In the above figure, PQRS is a square with side x + 4. Each of the [#permalink]

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20 Sep 2017, 10:26
souvonik2k wrote:
Hi Genxer123
Please check the red font portion marked above.
Area of the large square shall be $$(x+4)^2$$
and x is the length of the side of rectangular portion of shaded area.
Please refer my solution in the previous post.

souvonik2k

Shoot, I solved for the length of the side, not x! Must read more carefully. You are correct. I'll edit. Thanks, and kudos.

P.S. JMO, but when you "notify" someone by username, it works better if you put an @ before the name (no space between the symbol and the name), no punctuation near it (leave one space for apostrophe, e.g.), and the name must be exact.

Usually, but not always, I check topics on which I have posted, when email tells me a subsequent post exists. If email says my username was mentioned, I always check . . . Because it might just be an astute correction like yours. Cheers!
_________________

At the still point, there the dance is. -- T.S. Eliot
Formerly genxer123

In the above figure, PQRS is a square with side x + 4. Each of the   [#permalink] 20 Sep 2017, 10:26
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