Find two consecutive odd positive integers, sum of whose square is 290

1.	Find two consecutive odd positive integers, sum of whose square is 290.
Answer» Let x and x + 2 are two consecutive odd positive integers. According to question, (x)² + (x + 2)² = 290 ⇒ x² + x² + 4x + 4 = 290 ⇒ 2x² + 4x + 4 = 290 ⇒ 2x² + 4x – 286 = 0 ⇒ x² + 2x – 143 = 0 ⇒ x² + 13x – 11x – 143 = 0 ⇒ x(x + 13) – 11(x + 13) = 0 ⇒ (x + 13) (x – 11) = 0 ⇒ x – 11 = 0 ⇒ x = 11 Consecutive odd numbers x = 11 x + 2 = 11 + 2 = 13 Hence, two consecutive odd positive integers are 11, 13

Find two consecutive odd positive integers, sum of whose square is 290.

Answer»

Let x and x + 2 are two consecutive odd positive integers.

According to question,

(x)² + (x + 2)² = 290

⇒ x² + x² + 4x + 4 = 290

⇒ 2x² + 4x + 4 = 290

⇒ 2x² + 4x – 286 = 0

⇒ x² + 2x – 143 = 0

⇒ x² + 13x – 11x – 143 = 0

⇒ x(x + 13) – 11(x + 13) = 0

⇒ (x + 13) (x – 11) = 0

⇒ x – 11 = 0

⇒ x = 11

Consecutive odd numbers x = 11

x + 2 = 11 + 2 = 13

Hence, two consecutive odd positive integers are 11, 13