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Question 167137This question is from textbook Algebra and Trigonometry Structure and Method book 2
: I have been trying to figure out this math problem and I can't get the answer. I was wondering if someone could help me? Please and Thank You!! I would deeply appreciate it!!
Problem Solving Using Polynomial Equations
Solve each problem. If there are two correct answers, give both of them.
A rectangle is 15cm wide and 18cm long. If both dimensions are decreased by the same amount, the area of the new rectangle formed is 116cm^2 less than the area of the original. Find the dimensions of the new rectangle. This question is from textbook Algebra and Trigonometry Structure and Method book 2

Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
A rectangle is 15cm wide and 18cm long. If both dimensions are decreased by the same amount, the area of the new rectangle formed is 116cm^2 less than the area of the original. Find the dimensions of the new rectangle.
.
Let x = amount decreased
.
new width = 15-x
new length = 18-x
.
(15-x)(18-x) = 116
270-15x-18x+x^2 = 116
270-33x+x^2 = 116
154-33x+x^2 = 0
x^2-33x+154 = 0
.
Using the quadratic equation to solve we get:
x = {27.374, 5.626}
.
We can toss out the 27.374 leaving us with
x = 5.626
.
new width = 15-x = 15-5.626 = 9.374 inches
new length = 18-x = 18-5.626 = 12.374 inches
.
Solution: new rectangle is 9.374 by 12.374 inches
.
Details of quadratic:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-33x%2B154+=+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=%28-33%29%5E2-4%2A1%2A154=473.

Discriminant d=473 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--33%2B-sqrt%28+473+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-33%29%2Bsqrt%28+473+%29%29%2F2%5C1+=+27.3742815854658
x%5B2%5D+=+%28-%28-33%29-sqrt%28+473+%29%29%2F2%5C1+=+5.62571841453423

Quadratic expression 1x%5E2%2B-33x%2B154 can be factored:
1x%5E2%2B-33x%2B154+=+1%28x-27.3742815854658%29%2A%28x-5.62571841453423%29
Again, the answer is: 27.3742815854658, 5.62571841453423. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-33%2Ax%2B154+%29