Find the side lengths of a rectangle with an area of 51 cm2 and a perimeter of 40 cm.

Let x cm be the length of the rectangle and y be its width. Then the area of the rectangle is S = xy, and by condition 51. The perimeter of the rectangle is P = (x + y) 2, and by condition 40. Let’s compose and solve the system of equations.
{xy = 51
(x + y) 2 = 40.
We transform the second equation and express from it x
x + y = 20
x = 20 – y. Substitute in the first equation of the system.
(20-y) y = 51
20y – y ^ 2 = 51
-y ^ 2 + 20y-51 = 0
y ^ 2-20y + 51 = 0 by Vietta’s theorem
y1 + y2 = 20
y1 * y2 = 51
y1 = 3
y2 = 17.
Substitute the roots to x = 20-y and get
x1 = 17
x2 = 3.