SOLUTION: A rectangle has a diagonal of 15 cm. The width of the rectangle is 3 cm less than its lenght. Find the area of the rectangle.

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Question 101752: A rectangle has a diagonal of 15 cm. The width of the rectangle is 3 cm less than its lenght. Find the area of the rectangle.
Answer by doukungfoo(195) About Me  (Show Source):
You can put this solution on YOUR website!
Ok first lets sketch out what we know
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From the drawing we can see that the diagonal forms a right triangle.
So we can use the pythagorean formula to solve for x.
The pythagorean formula is a%5E2%2Bb%5E2=c%5E2 where c is the hypotenuse of a right triangle. a and b are the other two sides of a right triangle.
So lets start:
x%5E2%2B%28x-3%29%5E2=15%5E2
lets start on the right side and go ahead and square 15
x%5E2%2B%28x-3%29%5E2=225
now on the left side square (x-3) use the foil method to do this
x%5E2%2Bx%5E2-3x-3x%2B9=225
combine like terms
2x%5E2-6x%2B9=225
Ok the pythagorean formula got us this far but now we need to use the quadratic formula to continue solving for x. To do this we first need to set the the equation equal to zero. so...
2x%5E2-6x%2B9=225
subtract 225 from both sides
2x%5E2-6x-216=0
now we can use the quadratic formula to solve for x
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-6x%2B-216+=+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=%28-6%29%5E2-4%2A2%2A-216=1764.

Discriminant d=1764 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--6%2B-sqrt%28+1764+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+1764+%29%29%2F2%5C2+=+12
x%5B2%5D+=+%28-%28-6%29-sqrt%28+1764+%29%29%2F2%5C2+=+-9

Quadratic expression 2x%5E2%2B-6x%2B-216 can be factored:
2x%5E2%2B-6x%2B-216+=+2%28x-12%29%2A%28x--9%29
Again, the answer is: 12, -9. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-6%2Ax%2B-216+%29


Ok so when solving quadratic equations you get two answers.
the answers we got here are x = -9 and x = 12
They are both correct answers mathematically.
But for our problem we are trying to find the length and width of a rectangle and neither of these can be a negative number.
So we will throw out the x = -9 answer as an 'extraneous solution'
That leaves just x = 12
Ok back to the rectangle from the diagram we started with we can see that the:
Length = x and the width = x - 3
so
Length = 12 and the width = 12 - 3 = 9
Find area of rectangle by Length times width
12 * 9 = 108
So finally the area of the rectangle is 108cm%5E2