Find two consecutive natural numbers, the sum of whose squares is 145.

Let two consecutive natural numbers be x and x + 1.

Given,

The sum of squares of the numbers is 145.

⇒ x² + (x + 1)² = 145

⇒ x² + x² + 2x + 1 = 145

⇒ 2x² + 2x + 1 - 145 = 0

⇒ 2x² + 2x - 144 = 0

⇒ 2(x² + x - 72) = 0

⇒ x² + x - 72 = 0

⇒ x² + 9x - 8x - 72 = 0

⇒ x(x + 9) - 8(x + 9) = 0

⇒ (x - 8)(x + 9) = 0

⇒ (x - 8) = 0 or (x + 9) = 0     [Using zero-product rule]

⇒ x = 8 or x = -9.

Since, they are consecutive natural numbers, x ≠ -9.

Thus, x = 8 and x + 1 = 9.

Hence, two consecutive natural numbers 8 and 9.