The area of the rectangle is 36 cm ^ 2, and its perimeter is 24 cm. Find the sides.

The perimeter and area of ​​the rectangle are known. Let the length of the rectangle be x cm and the width y cm.Let’s make a system of equations:

x * y = 36;
x + y = 24/2 = 12.

Let us express from the second equation x

x = 12 – y;

Substitute the value of the variable x into the first equation of the system:

(12 – y) * y = 36;
12y – y ^ 2 = 36;
y ^ 2 – 12y + 36 = 0;

Find the discriminant of the quadratic equation:

D = b ^ 2 – 4 * a * c = (-12) ^ 2 – 4 * 1 * 36 = 144 – 144 = 0

Since the discriminant is zero, the quadratic equation has one real root:

x = 12/2 * 1 = 6 (cm) – side of the rectangle.

Because there is only one root, this means that the width is equal to the length, and the given rectangle is a square.