A rectangle, one side of which is 2 cm larger than the other, has an area equal to the area of a square

A rectangle, one side of which is 2 cm larger than the other, has an area equal to the area of a square with a side 4 cm less than the perimeter of the rectangle. find the sides of the rectangle.

Let us introduce the notation. Let a be one side of the rectangle. Then the second side will be (a + 2).

Let us express the area of ​​the rectangle: S = a (a + 2) = a ^ 2 + 2a.

Let us express the perimeter of the rectangle: P = (a + a + 2) * 2 = (2a + 2) * 2 = 4a + 4.

The side of the square is 4 cm less than the perimeter of the rectangle, we express the side of the square: 4a + 4 – 4 = 4a.

Let us express the area of ​​this square: 4a * 4a = 16a ^ 2.

The areas of the rectangle and the square are equal, let’s make the equation:

a ^ 2 + 2a = 16a ^ 2;

16a ^ 2 – a ^ 2 – 2a = 0;

15a ^ 2 – 2a = 0;

a (15a – 2) = 0.

a = 0 (does not satisfy the condition).

15a – 2 = 0; 15a = 2; a = 2/15 cm.

2 + 2/15 = 2 2/15 cm.

Answer: the sides of the rectangle are 2/15 cm and 2 2/15 cm.