What can be the lengths of the sides of a rectangle, the perimeter of which is 26 cm, and the area is 40 cm?

Suppose x is the length of the rectangle and y is the width. Then the perimeter is P = (x + y) × 2 and the area S = xy. Let us compose a symtem of equations
{(x + y) 2 = 26
xy = 40.
Divide the first equation by 2 (x + y = 13) and express x
x = 13-y
Substitute x in the second equation
(13-y) y = 40
13y-y ^ 2-40 = 0 divided by -1
y ^ 2-13y + 40 = 0
By Vieta’s theorem, we find the roots
y1 + y2 = 13
y1 × y2 = 40
y1 = 5 y2 = 8
Substitute the roots in the equation x = 13-y
x1 = 8
x2 = 5
Answer: the length of the rectangle is 8 and the width is 5, or another option is the length is 5 and the width is 8.