SOLUTION: 52. A rectangular painting has dimensions x and x + 10. The painting is in a frame 2 in. wide. The total area of the picture and the frame is 900 in.

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: 52. A rectangular painting has dimensions x and x + 10. The painting is in a frame 2 in. wide. The total area of the picture and the frame is 900 in.       Log On


   

Question 575668: 52. A rectangular painting has dimensions x and x + 10. The painting is in a
frame 2 in. wide. The total area of the picture and the frame is 900 in.
Answer by JBarnum(2146) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%2B4%29%28x%2B14%29=900
how did i get this equation?
the frame is 2 inches thick so theres 2 inches on left side of the length and 2 inches on the right side so thats 4 more inches than the lenth of the picture
same goes for the width 2 in on top 2 inches on bottom makes 4 more inches added to width
LW=A length times width equals area
A=900
L=x%2B4
W=%28x%2B10%29%2B4
W=x%2B14
back to the equation
%28x%2B4%29%28x%2B14%29=900 FOIL (First Outer Inner Last)
F x%2Ax=x%5E2
O x%2A14=14x
I 4%2Ax=4x
L 4%2A14=56
x%5E2%2B14x%2B4x%2B56=900 add like terms
x%5E2%2B18x%2B56=900subtract 900 from both sides
x%5E2%2B18x-844=0 use quadratic formula to solve
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B18x%2B-844+=+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=%2818%29%5E2-4%2A1%2A-844=3700.

Discriminant d=3700 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-18%2B-sqrt%28+3700+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2818%29%2Bsqrt%28+3700+%29%29%2F2%5C1+=+21.4138126514911
x%5B2%5D+=+%28-%2818%29-sqrt%28+3700+%29%29%2F2%5C1+=+-39.4138126514911

Quadratic expression 1x%5E2%2B18x%2B-844 can be factored:
1x%5E2%2B18x%2B-844+=+1%28x-21.4138126514911%29%2A%28x--39.4138126514911%29
Again, the answer is: 21.4138126514911, -39.4138126514911. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B18%2Ax%2B-844+%29