Question

Find three consecutive positive odd integers, the sum of whose squares is 155.

Answer 1

Let the three consecutive positive odd integers be: x, x + 2, x + 4

Given,

The sum of squares of three consecutive positive odd integers is 155.

⇒ x² + (x + 2)² + (x + 4)² = 155

⇒ x² + x² + 4x + 4 + x² + 8x + 16 = 155

⇒ 3x² + 12x + 20 = 155

⇒ 3x² + 12x + 20 − 155 = 0

⇒ 3x² + 12x − 135 = 0

⇒ 3(x² + 4x − 45) = 0

⇒ x² + 4x − 45 = 0

⇒ x² + 9x − 5x − 45 = 0

⇒ x(x + 9) − 5(x + 9) = 0

⇒ (x − 5)(x + 9) = 0

⇒ (x − 5) = 0 or (x + 9) = 0

⇒ x = 5 or x = -9.

Since the integers are positive, x = 5

The three consecutive positive odd integers,

x = 5,

x + 2 = 5 + 2 = 7,

x + 4 = 5 + 4 = 9.

Hence, three consecutive positive odd integers are 5, 7, 9.