SOLUTION: Suppose the length of one rectangular field is 400 meters more than the side of a square field. The width of the rectangular field is 100 meters more than the side of the square fi

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Question 542362: Suppose the length of one rectangular field is 400 meters more than the side of a square field. The width of the rectangular field is 100 meters more than the side of the square field. If the rectangular field has twice the area of the square field, what are the dimensions of each field?
I came up with 2A^2=(A+400)(A+100)
A being equal to the side of the square, 2A^2 being equal to the squares area
I'm not sure where to go from there because when I foiled it out to:
2A^2=A^2+500A+40,000 I am not sure how to solve for A or if that is even the correct variable.
Answer by neatmath(302) About Me  (Show Source):
You can put this solution on YOUR website!

You have 3 possible variables to work with:

l length of rect field

w width of rect field

s side of square field

We know:

l=400%2Bs

w=100%2Bs

l%2Aw=2%2As%5E2

Now let's substitute those first 2 values for l and w into our 3rd equation:

%28400%2Bs%29%2A%28100%2Bs%29=2%2As%5E2

40000%2B100s%2B400s%2Bs%5E2=2%2As%5E2

40000%2B500s%2Bs%5E2=2%2As%5E2

s%5E2-500s-40000=0

Then we can factor this or use the quadratic formula:

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-500x%2B-40000+=+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=%28-500%29%5E2-4%2A1%2A-40000=410000.

Discriminant d=410000 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--500%2B-sqrt%28+410000+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-500%29%2Bsqrt%28+410000+%29%29%2F2%5C1+=+570.156211871642
x%5B2%5D+=+%28-%28-500%29-sqrt%28+410000+%29%29%2F2%5C1+=+-70.1562118716424

Quadratic expression 1x%5E2%2B-500x%2B-40000 can be factored:
1x%5E2%2B-500x%2B-40000+=+%28x-570.156211871642%29%2A%28x--70.1562118716424%29
Again, the answer is: 570.156211871642, -70.1562118716424. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-500%2Ax%2B-40000+%29


So your equation was right, you just needed to go a bit further.

After using the quadratic formula, we can see there is only one possible answer for s:

s=570.156 meters rounded to 3 decimal places

We had to discard the other possible solution because it was negative, and we can't have negative distances (in most cases).
And once we know the value for s, we can easily calculate the values for l and w.

l=970.156 meters

w=670.156 meters

I hope this helps!