Find the area of the orginal square.

Question

If two opposite sides of a square are increased by 13 meters and the other sides are decreased by 6 meters, the area of the rectangle that is formed is 66 square meters. Find the area of the original square.

Answer 1

A square has 4 sides that are equal to each other.

 

Let   x = side

 

 

New Rectangle:

 

Two sides = x + 13

The other sides = x - 6

 

Then the area of this rectangle is

 

(x + 13)(x - 6) = 66

 

x² + 7x - 78 = 66

 

 

Subtract 66 on both sides of the equation.

 

x² + 7x - 144 = 0

 

 

Use the quadratic formula to solve for x from this quadratic equation.  Be sure to choose the positive value of x.  Then once you found x, square that value to get the area of the original square.

 

 

Answer 2

 

 

 

Let's assign the following:

 

s..............side of square

L=s+13....one side of rectangle        

                (side of square 

                 increased by 13)

W=s-6....other side of

              rectangle (side of

              square decreased

              by 6)

 

A_rect=LxW..................area of rectangle

     66=(s+13)(s-6)......substitute

     66=x²+7x-78.........expand (FOIL)

     x²+7x-144=0.........combine like terms,

                                     (Standard Form

                                     Ax²+Bx+C=0)

     a=1, b=7, c=-144...coefficients for the

                                     Quadratic Formula

     

   s= [-b ± √(b²-4ac)]/(2a)...Quadratic Formula

     = [-7 ± √((7²-(4)(1)(-144))]/((2)(1))...

                                           substitute for

                                           variables "a, b, c"

     = [-7 ± √(49+576)]/2......simplify

     = (-7 ± √625)/2

     = (-7 ± 25)/2

     = (-7 +25)/2....evaluate positive root

     = 18/2

∴ s= 9

 

     -OR-

 

   s= (-7-25)/2......evaluate negative root

     = -32/2

∴ s= -16..............disregard root, distance

                               cannot be negative

      

Substitute "s=9" into assigned rectangle dimensions... 

 L=9+13=22 meters

W=9-6=3 meters

A_rect=LxW

   66=22x6

   66=66..............true, √check, our solution

                            is correct. (always check

                            your work)

∴ A_square=s²=9²=81 square meters

 

...and the rectangle dimensions are 22 meters by 3 meters, the square dimensions are 9 meters by 9 meters.