# One side of the rectangle is 10 cm smaller than the other. If the smaller side is increased by 15 cm and the larger

**One side of the rectangle is 10 cm smaller than the other. If the smaller side is increased by 15 cm and the larger one is increased by 20 cm, then the area of the rectangle will increase by 5 times. What is the width of this rectangle?**

Suppose that the large side of this rectangle is x cm, then the smaller side will be x – 10 cm.

The area of this rectangle is:

x * (x – 10) = x² – 10 * x (cm²).

According to the condition of the problem, the smaller side was increased by 15 cm, which means its size turned out to be x – 10 + 15 = x + 5 cm. The larger side was increased by 20 cm, which means its size became x + 20 cm.

The area of the resulting rectangle is:

(x + 5) * (x + 20) = x² + 25 * x + 100 (cm²).

Let’s compose and solve the equation:

5 * (x² – 10 * x) = x² + 25 * x + 100,

4 * x² – 75 * x – 100 = 0.

The discriminant of this equation is:

(-75) ² – 4 * 4 * (-100) = 7225.

Since x can only be a positive number, the problem has a unique solution:

x = (75 + 85) / 8 = 20 (cm).

So the second side of the rectangle is 20 – 10 = 10 (cm).