Solving a word problem using a quadratic equation with rational roots

Question

The area of a rectangle is 54ft^2, and the length of the rectangle is 3ft more than twice the width. Find the dimensions of the rectangle.

Length: ____ ft
Width: ____ ft

Answer 1

So to solve this, let's start with standard definitions. For a rectangle, A=base x height and Perimeter = 2w+2L

We know that L=2w+3 from the problem and since Area is 54 square ft, then L*w=54, therefore...

54=(2w+3)*(w)
54=2w^2+3w
2w^2+3w-54=0

Now you have your quadratic equation!
Since you know it has rational roots, you can attempt to solve by factoring.

(w+6)(2w-9)=0. Next you separate the factors,
w+6=0    2w-9=0
    w=-6        w=(9/2)

Since we are looking for the lengths of a rectangle, our answer cannot be negative, so we through away (w=-6) and just keep w=(9/2)

So to find length, let's plug it back into the original L equation
L=2w+3
L=2(9/2)+3
L=9+3
L=12

Hope this helps! Let me know if you have any more questions or any follow-up that I can help you with!
Nathaniel