One side of a rectangle with an area of 189 cm. In a square is 12 cm larger than the other side.

One side of a rectangle with an area of 189 cm. In a square is 12 cm larger than the other side. Make an equation, denoting through X: a) the smaller side of the rectangle: b) the larger side of the rectangle

Part 1.

We denote by x the length of the larger side of the rectangle, then the second side is equal to (x – 12) cm.

The area of ​​the rectangle is found by the formula: S = a x b, where S is the area, a is the length, and b is the width.

Thus, we get:

x ∙ (x – 12) = 189.

x2 – 12x = 189.

x2 – 12x – 189 = 0 is a quadratic equation.

We find the discriminant:

D = (-12) 2 – 4 x 1 x (-189) = 144 + 756 = 900.

D is greater than zero, so the equation has 2 roots.

x1 = (- (-12) + √900) / 2 = (12 + 30) / 2 = 42/2 = 21.

x2 = (- (-12) – √900) / 2 = (12 – 30) / 2 = -18 / 2 = -9 – this root does not fit, because x is the length of the side of the rectangle, which cannot be expressed as a negative number.

So, the length of the rectangle is 21 cm, and its width is: 21 – 12 = 9 (cm).

Part 2.

If we denote the length of the smaller side for x, then the larger side will be: (x + 12) cm.

The equation will be like this:

x ∙ (x + 12) = 189.

D = 900.

x1 = 9.

x2 = -21.

The smaller side is 9 cm and the larger side is: 9 + 12 = 21 (cm).