# The area of the rectangle is 30 cm2 and the perimeter is 26 cm, what is the length of its sides?

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

Let’s write down what the perimeter of this rectangle is.

2x + 2y = 26.

Let’s write down what is the area of ​​the rectangle.

x * y = 30.

Let’s compose a system of equations and solve it.

2x + 2y = 26,

x * y = 30.

Divide the first equation by 2.

x + y = 13,

x * y = 30.

Find x from the first equation.

x = 13 – y,

x * y = 30.

Substitute the first equation into the second.

(13 – y) * y = 30.

13y – y ^ 2 = 30.

13y – y ^ 2 – 30 = 0.

-y ^ 2 + 13y – 30 = 0.

We got a quadratic equation, multiply both sides by (- 1) and solve it.

y ^ 2 – 13y + 30 = 0.

D = b ^ 2 – 4ac = 169 – 4 * 1 * 30 = 49> 0.

y1 = (13 – 7): 2 * 1 = 3.

y2 = (13 + 7): 2 * 1 = 10.

x1 = 13 – y1 = 13 – 3.

x1 = 10

x2 = 13 – y2 = 13 – 10.

x2 = 3.

Hence, the sides of the rectangle are 3 cm and 10 cm, or 10 cm and 3 cm.

Answer: The sides of the rectangle are 3 cm and 10 cm, or 10 cm and 3 cm.

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