I don't understand this type of problems

Question

wesley just received an award for employee of the year and wants to create a border of uniform width around the rectangular plaque. He only has enough material to create a total of 88 square inches in area for the border. If the plaque is 8 inches x 10 inches, how wide should the border be?

Answer 1

First, draw a picture -- the plaque is like a picture that has a frame around it.  The frame is "uniform width" around the picture.

 

Let x=the width of the border.  That means that the new width is (x+8+x) and the new height is (x+10+x).

 

The amount of material that Wesley can use is 88 square inches.  That is the size with the border minus the size of the plaque.  So:

     (2x+8)*(2x+10) - 8*10 = 88

 

Now, you can solve for x, "how wide should the border be?"

 

   4x² +20x + 16x + 80   -  80 = 88       [Use F-O-I-L to expand the expression]

   4x² + 36x - 88 = 0                 [subtract 88 from both sides]

    x² + 9x - 22 = 0         [divide everything by 4 to make it easier]

   (x + 11)*(x - 2) = 0        [either factor or use quadratic formula]

  Either x=-11 or x=2        [from the multiplicative property of zero,

                                at least one of the factors must be zero]

     x = 2         [frame width cannot be negative]

 

Check:

  Is  (2+8+2)*(2+10+2)  -  8*10  = 88   ?

       12*14 - 80  =  88   ?

         168 - 80 = 88   ?

            88 = 88   ?yes