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.
