Find the sides of a rectangle if its width is 6 cm less than its length and its area is 40 cm ^ 2.

1. Let the length of the rectangle be x cm. It is known that the width is 6 cm less than the length, which means that the width of the rectangle is (x – 6) cm.

2. The area of ​​a rectangle is equal to the product of its length and width. Then we can write the equality:

x * (x – 6) = 40;

x ^ 2 – 6x = 40;

x ^ 2 – 6x – 40 = 0;

3. Let’s solve the resulting quadratic equation. First, we find the discriminant:

D = b ^ 2 – 4 * a * c = (-6) ^ 2 – 4 * 1 * (-40) = 36 + 160 = 196.

Because the discriminant is greater than zero, the quadratic equation has two real roots:

x1 = (6 – √196) / 2 = (6 – 14) / 2 = -8 / 2 = -4;

x2 = (6 + √196) / 2 = (6 + 14) / 2 = 20/2 = 10;

4. Obviously, the length of the rectangle is greater than 0, which means that the length of the rectangle is 10 cm, and the width is 10 – 6 = 4 cm.

Answer: the length of the rectangle is 10 cm, the width of the rectangle is 4 cm.