What are the sides of a rectangle whose area is 12 cm and the perimeter is 16 cm?

Let’s write down the formulas for the area and perimeter of the rectangle.

S = ab and P = 2 (a + b).

Semi-perimeter P / 2 = a + b, a + b = 16: 2 = 8.

Let one side of the rectangle be x, then the other is (8 – x).

Equation using the area formula, x (8 – x) = 12.

8x – x ^ 2 – 12 = 0,

– x ^ 2 + 8x – 12 = 0,

x ^ 2 – 8x + 12 = 0.

We solve the quadratic equation by making a system of the sum and the product of the roots.

x1 + x2 = 8 and x1 * x2 = 12.

We select the roots so that the system conditions are met.

x1 = 2, x2 = 6.

Answer: 2 cm and 6 cm sides of the rectangle.