SOLUTION: The length of each side of a square is increased by 4 in. The sum of the areas of the original square and the larger square is 106 in^2. What is the length of a side of the origina

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Question 214941: The length of each side of a square is increased by 4 in. The sum of the areas of the original square and the larger square is 106 in^2. What is the length of a side of the original square?
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
The length of each side of a square is increased by 4 in. The sum of the areas of the original square and the larger square is 106 in^2. What is the length of a side of the original square?

Step 1. Let s be the length of the side of the original square then Area A of a square is A=s%5E2.

Step 2. %28s%2B4%29%5E2 Area of square with side increased by 4.

Step 3. Then using the problem statement where the sum of the areas of the two squares is 106 square inches, we have

s%5E2%2B%28s%2B4%29%5E2=106

{{2{s^2+8s+16=106}}}

Step 4. Subtract 106 from both sides to get a quadratic equation

2s%5E2%2B8s%2B16-106=106-106

2s%5E2%2B8s-90=0

Step 5. We can now use the quadratic formula to solve this equation given as

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=2, b=8 and c=-90

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation as%5E2%2Bbs%2Bc=0 (in our case 2s%5E2%2B8s%2B-90+=+0) has the following solutons:

s%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%288%29%5E2-4%2A2%2A-90=784.

Discriminant d=784 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-8%2B-sqrt%28+784+%29%29%2F2%5Ca.

s%5B1%5D+=+%28-%288%29%2Bsqrt%28+784+%29%29%2F2%5C2+=+5
s%5B2%5D+=+%28-%288%29-sqrt%28+784+%29%29%2F2%5C2+=+-9

Quadratic expression 2s%5E2%2B8s%2B-90 can be factored:
2s%5E2%2B8s%2B-90+=+2%28s-5%29%2A%28s--9%29
Again, the answer is: 5, -9. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B8%2Ax%2B-90+%29



Select the positive solution of s=5.

As a check, s%5E2%2B%28s%2B4%29%5E2=+25%2B81=106. So it works!

Step 6. The length of the original square is 5 inches.

I hope the above steps were helpful.

For FREE Step-By-Step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

Good luck in your studies!

Respectfully,
Dr J