The area of a rectangle is equal to 132 square meters. the sum of two of the perpendicular sides of the rectangle is equal to 23 meters. what are the measures of each of these sides

Let the length rectangle be x, and breadth be y. These are the two perpendicular sides.

   x*y = 132    (a)

   x  + y = 23. (b)                     

(From the top of your head you know the answers are 12 and 11, because 12 * 11 = 132, and 12 + 11 = 23, but let us just go ahead and solve it)

From equation (b):

  x = 23 - y(c).

Substituting (c) in (a).

x*y  = 132

(23 - y)y = 132
 
23y - y² = 132

0 = 132 + y² - 23y

0 = y² - 23y + 132.        

This is a quadratic equation and we solve by factorization. 

y² - 23y + 132 = 0.   

This can be solved directly in a Calculator that quadratic function is already programmed, without having to go through the workings below.

y² - 23y + 132 = 0

Multiply first and last coefficients:  1*132 = 132

We look for two numbers that multiply to give 132, and add to give the middle term -23

Those two numbers are -11 and -12.

Check:   -11*-12 = 132         -11 + -12 = -11 - 12 = -23

We replace the middle term of -23y in the quadratic expression with -11y -12y

 y² - 23y + 132 = 0     

y² - 11y - 12y + 132 = 0     

y(y - 11) - 12(y - 11) = 0

(y - 11)(y - 12) = 0

y - 11 = 0    or   y - 12 = 0

y = 0 +11              y = 0 + 12

y = 11                    y = 12

y = 11 or 12.

Recall (c)

  x = 23 - y(c).

When y = 11,  x = 23 - y = 23 - 11 = 12

When y = 12,  x = 23 - y = 23 - 12 = 11

The pair of the solution is (x, y) =  (12, 11)  or (11, 12)

We would take only one pair of solution.

Therefore the length  and breadth of rectangle are 12 and 11 meters.