SOLUTION: A picture 19 inches by 24 inches is to be mounted on a piece of matboard so that there is an even amount of mat all around the picture. How wide will the border be if the area of t

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Question 301267: A picture 19 inches by 24 inches is to be mounted on a piece of matboard so that there is an even amount of mat all around the picture. How wide will the border be if the area of the border is 188 square inches?
Answer by nerdybill(7384) (Show Source):
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A picture 19 inches by 24 inches is to be mounted on a piece of matboard so that there is an even amount of mat all around the picture. How wide will the border be if the area of the border is 188 square inches?
.
Let w = width of border
then
"area of border" = "area of mat" - "area of picture"
188 = (19+2w)(24+2w) - 19(24)
188 = 456+38w+48w+4w^2 - 456
188 = 38w+48w+4w^2
188 = 4w^2+86w
0 = 4w^2+86w-188
0 = 2w^2+43w-94
Solve using the quadratic formula. Doing so yields:
w = {2, -23.5}
Of course, we can toss out the negative solution leaving:
w = 2 inches
.
Details of quadratic:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=2601 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 2, -23.5. Here's your graph: