SOLUTION: The length of a rectangular picture is 5 cm more than the width. The area is 84 cm^2. Find the length and width

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Question 357105: The length of a rectangular picture is 5 cm more than the width. The area is 84 cm^2. Find the length and width
Answer by neatmath(302) About Me  (Show Source):
You can put this solution on YOUR website!
Let l be the length. Let w be the width.

Thus, we know:

l%2Aw=84 and l=w%2B5

We can use this information to solve:

%28w%2B5%29%2Aw=84

w%5E2%2B5w=84

w%5E2%2B5w-84=0

We then need to factor this equation, or use the quadratic formula, like this:

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B5x%2B-84+=+0) has the following solutons:

x%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=%285%29%5E2-4%2A1%2A-84=361.

Discriminant d=361 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-5%2B-sqrt%28+361+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%285%29%2Bsqrt%28+361+%29%29%2F2%5C1+=+7
x%5B2%5D+=+%28-%285%29-sqrt%28+361+%29%29%2F2%5C1+=+-12

Quadratic expression 1x%5E2%2B5x%2B-84 can be factored:
1x%5E2%2B5x%2B-84+=+1%28x-7%29%2A%28x--12%29
Again, the answer is: 7, -12. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B5%2Ax%2B-84+%29


Since the width has to be a positive number, we know that the width is 7, and we can discard the -12 solution.

Then, we have:

l=w%2B5

l=7%2B5

l=12

Then, our answer is: the length is 12 cm, and the width is 7 cm.

I hope this helps!