Steve's backyard measures 75 ft by 50 ft. because of his spies and enemies he builds a moat of uniform width to surround the backyard. the moat has decreased the area of his backyard by 425 square feet. calculate the width of the moat

Original area of backyard:  (75 ft)(50 ft) = 3750 sq ft.

Less moat area (425 sq ft):                      -  425 sq ft
                                                               
                                                                    3325 sq ft

Let the new yard length be L and width W.  Then L*W = 3325 sq ft.

If the moat width is x feet, then L=75 ft   -   x, and W = 50 ft - x.

Since L*W = 3325 sq ft  =  L*W,   (75 ft) - x   times   (50 ft - x) = 3325 sq ft.

Then 75*50 - 75x - 50x + x^2 = 3325.  Rewriting this in std. form,

x^2 -125x + 3750 - 3325 = 0, or x^2 - 125x + 425 = 0

We must solve this for x (the moat width).

Using the quadratic formula:
          125 plus or minus sqrt(125^2 - 4(1)(425) )
 x =
                                         2
         125 plus or minus sqrt(13925)            125 plus or minus 118
 x =  =
                                 2                                                   2

This results in 2 positive answers.  We will need to find them, and then decide which one is the reasonable choice for this real-life situation.

125+118
 =  121.5 feet  (this is ridiculous, since the original length of the0
       2                               yard was 75 feet.  Reject x = 121.5.

125-118
= 7/2, or 3 1/2 feet.
      2

That's not a very wide moat, but it's a lot more reasonable than 121.5 ft as moat width!

The width is 3000000000