What are the sides of a rectangle if its perimeter is 24 and its area is 20?

For a rectangle, the opposite sides are equal and parallel and are called length (a) and width (b).
Since the perimeter of any polygon is equal to the sum of the lengths of all its sides, the perimeter of the rectangle is:

P = a + b + a + b = 2 * a + 2 * b = 2 * (a + b).

The area of ​​a rectangle is equal to the product of its length and width, then:
S = a * b.
Substitute the data on the value condition in the formulas and solve the system of linear equations with two variables:
2 * (a + b) = 24;
a * b = 20.
In the second equation of the system, we express the variable a:
a = 20 / b.
Substitute this expression into the first equation and solve the quadratic equation in one variable:
2 * (20 / b + b) = 24;
2 * (20 / b + b ^ 2 / b) = 24;
2 * (20 + b ^ 2) / b = 24;
2 * (20 + b ^ 2) = 24 * b;
40 + 2 * b ^ 2 – 24 * b = 0;
b ^ 2 – 12 * b + 20 = 0.
Let’s find the discriminant:
D = 12 ^ 2 – 4 * 1 * 20 = 144 – 80 = 64.
b1 = (12 – 8) / 2 = 4/2 = 2;
b2 = (12 + 8) / 2 = 20/2 = 10.
Find the length value:
a1 = 20 / b1 = 20/2 = 10;
a2 = 20 / b2 = 20/10 = 2.
Answer: the sides of a rectangle with a perimeter of 24 and an area of ​​20 are equal to 2 and 10 units of length.