The perimeter of the rectangle is 30cm, and its area is 54cm2. Find the length and width

As you know, the perimeter of a rectangle is calculated by the formula: P = (a + b) * 2, where a is the length of the rectangle, b is the width of the rectangle.
Substitute all known values ​​in the formula: 30 = (a + b) * 2.
As you know, the area of ​​a rectangle is calculated by the formula: S = a * b, where a is the length of the rectangle, b is the width of the rectangle.
Substitute all known values ​​into the formula: 54 = a * b.
a = 54 / b.
Substitute instead of a into the perimeter formula and find side b: (54 / b + b) * 2 = 30.
108 / b + 2b = 30.
108 + 2b² = 30b.
2b² – 30b + 108 = 0.
D = b² – 4ac = 900 – 4 * 2 * 108 = 900 – 864 = 36.
x1 = (-b + √D) / 2a = (30 + 6) / 2 * 2 = 36/4 = 9.
x2 = (-b – √D) / 2a = (30 – 6) / 2 * 2 = 24/4 = 6.
Side b is either 9 cm or 6 cm.
Find the side a = 54/9 = 6 cm.
Let’s check: 6 * 9 = 54. (6 + 9) * 2 = 30.
Answer: The length of the rectangle is 9 cm, the width is 6 cm.