The perimeter of the rectangle is 12m, and its area is 8m2. Find the sides of the rectangle.

1. Let’s denote the sides of the required rectangle by x and y.

2. The perimeter of the rectangle is equal to the sum of the four sides:

2x + 2y = 12;
2 (x + y) = 12;
x + y = 6. (1)
3. The area of a rectangle is equal to the product of two sides:

xy = 8. (2)

4. Let’s compose the system of equations (1) and (2):

{x + y = 6;
{xy = 8.

5. By Vieta’s converse theorem, x and y are the roots of the reduced quadratic equation:

p ^ 2 – 6x + 8 = 0;
D / 4 = 3 ^ 2 – 8 = 9 – 8 = 1;
p = 3 ± √1 = 3 ± 1;
p1 = 3 – 1 = 2;
p2 = 3 + 1 = 4.
Answer: 2 m and 4 m.