Problem Solving

In the figure, square LMNO has a side of length 2x + 1 and the two smaller squares have sides of length 3 and 6. If the area of the shaded region is 76, what is the value of x ? (please note the shaded region is not inside of the two small squares)

A). 5
B). 6
C). 7
D). 11
E). 14

OA is A

[spoiler]This question is from kaplan. The explanation says that area of square LMNO equals the sum of the shaded area and the area of two small squares. So, after performing some calculations an equation is formed:-

(2x + 1)² = 121
Shortest method now, which i completely agree
2x = 10
x = 5 (ans)

But i took a turn which was long and i thought that i will get the same answer; unfortunately i didn't. Can you please tell me where i am making a mistake in the following method ?
Actually, i solved the quadratic equation. It gives me 6 as the answer.

121 = 4x² + 1 + 4x
0 = 4x² - 4x - 120
0 = x² - x - 30
0 = x² - 6x + 5x - 30
0 = x(x - 6) + 5 (x - 6)
so, x = 6, x = -5
I am getting 6 here.[/spoiler]

Thanks
Sachin

Quote: 0 = 4x² - 4x - 120

This should instead be 0 = 4x²+ 4x - 120
solving this gives (x+6)*(x-5)=0
or x = 5 (since x cannot be -6)

Therefore x = 5

srcc25anu wrote:Quote: 0 = 4x² - 4x - 120

This should instead be 0 = 4x²+ 4x - 120
solving this gives (x+6)*(x-5)=0
or x = 5 (since x cannot be -6)

Therefore x = 5

My bad. I have done a very big careless mistake, and it must not have been done. Moreover, i could not believe myself after reading your above post.

Thanks

sachin_yadav wrote:In the figure, square LMNO has a side of length 2x + 1 and the two smaller squares have sides of length 3 and 6. If the area of the shaded region is 76, what is the value of x ? (please note the shaded region is not inside of the two small squares)

A). 5
B). 6
C). 7
D). 11
E). 14

Total area = smaller inner square + larger inner square + shaded area = 3² + 6² + 76 = 121.

In a square, area = (side)².
Since each side = 2x+1, we get:
121 = (2x + 1)²
11 = 2x + 1
x = 5.

The correct answer is A.

Thanks Mitch

GMATGuruNY wrote:
Total area = smaller inner square + larger inner square + shaded area = 3² + 6² + 76 = 121.

In a square, area = (side)².
Since each side = 2x+1, we get:
121 = (2x + 1)²
11 = 2x + 1
x = 5.

The correct answer is A.