The area of a rectangle is represented by the expression 2x^2 - 3x - 20in^2. I have to find the dimensions of the rectangle, include units, and leave my answers as binomials.

Question

If you can, please thoroughly explain what you are doing step by step, because I don't know how to do this problem. Thank you so much.

Answer 1

2x^2 - 3x - 20in^2

Setup:

1. The area of a rectangle is Width x Length, so with polynomials the solution will look something like this:
2. (aX +/- b) (cX +/- d)..... a,b,c,d = some number or integer, X= unknown variable
3. Watch this video as a refresher. The 3rd example in the video is an excellent example of how to factor a trinomial.

Steps:

1. (2x +/- ?)(x +/- ?)
2. 20 can be broken into the following: 1x20, 2x10, 4x5, or 2x2x5
3. We need a combination of two numbers, so this rules out 2x2x5.
4. We need one of the values to be multiplied by 2 because of (2x +/- ?)(x +/- ?)
5. We will use the 4x5 option because (4 times 2) - 5 = 3, which is our middle number from the original problem.
6. So, now we know more about the width and length of our rectangle: (2x +/- 5)(x +/- 4)
7. Now we need to determine if each binomial should include a + or a -
8. 2x²-3x-20... the 20 has a - preceding it. This means that EITHER the 5 or the 4 above must have a - preceding it, but not both.

I think you can take it from here.

SPOILER ALERT (Answer below)

(2x + 5)(x-4). This is your width and length of the rectangle... to include the in² it would be something like (2x + 5in²)(x-4in²)

Answer 2

There are an infinite number of factors that would work, but the obvious choice is factors with integer components - the standard factoring. A shortcut: the middle term is odd, so we know that the 2x term is going to be paired with an odd factor of 20.

(2x +5)(x - 4) clearly works.

The length is 2x + 5 in.

The width is x - 4 in.