The perimeter of the rectangle is 26 cm and its area is 42 cm². Find the sides of the rectangle.

Consider a rectangle with width a and length b. Since this is a rectangle, the formula for its perimeter (P) is:

P = 2 * a + 2 * b = 2 (a + b)
And the formula for the area (S) of a rectangle is:
S = a * b
Let us express the value of side a from the perimeter:

2 * (a + b) = 26
a + b = 13
a = 13-b
Substitute in the area formula:

(13-b) * b = 42
Let’s transform this expression:

Expand the brackets: 13 * b-b ^ 2 = 42
Then
-b ^ 2 + 13 * b-42 = 0 (* -1)

b ^ 2-13 * b + 42 = 0

we got the usual quadratic equation x ^ 2 – 13 * x + 42 = 0 (x = b), the solution of which will give us the value of one side of the rectangle.

Find the discriminant of the quadratic equation:

D = b ^ 2 – 4 * a * c = (-13) ^ 2 – 4 * 1 * 42 = 169 – 168 = 1

Since the discriminant is greater than zero, the quadratic equation has two real roots:

x1 = (13 – √1) / (2 * 1) = (13 – 1) / 2 = 12/2 = 6
x2 = (13 + √1) / (2 * 1) = (13 + 1) / 2 = 14/2 = 7

but then we get that our problem has two answers:
a = 13-b = 13-6 = 7
a = 13-b = 13-7 = 6

That is, a rectangle can be with sides a = 7cm, b = 6cm or a = 6cm, b = 7cm

Answer: the sides of the rectangle are 6cm and 7cm.