tofubeans wrote:So in this problem I am asked to find the area of a square within a rectangle with some givens.

The equation I come out with is: x^2+12x-540=0 and all I would have to do is solve for x. However, looking at that seems a little time confusing if I don't already know what numbers to choose (30 and 18).

Is there a method to this that doesn't require so much time in looking for the correct numbers that multiply out to 540 and subtract to 12? Or would I just have to take it as it is and test out a bunch of numbers until I hit the mark?

So, we know we're essentially looking for 2 numbers such that the product is -540 and the sum is 12.
Or we might just say that the product is 540 and the difference is 12.

Okay, let's start with 10 and 54. (10)(54)=540, but the difference is 44 (no good).

Aside: If (10)(54)=540, we can double one value and halve the other value and the product will still be 540. Double 10 to get 20, and halve 54 to get 27.

So, we know that (20)(27)=540, but the difference is 7 (no good).

Let's go back to (10)(54)=540. Triple 10 to get 30, and find the third of 54 to get 18.
So, we know that (30)(18)=540, and the difference is 18 (bingo!).

Factor x^2+12x-540 to get (x+30)(x-18)=0 . . . etc.

Cheers,
Brent