The sum of the squares of 2 natural consecutive odd numbers is 394. The sum of the numbers is :

Let, the two natural consecutive odd numbers be n and (n + 2)
Now, according to the question,
⇒ n² + ( n + 2 )² = 394
⇒ n² + n² + 4 + 4n = 394
⇒ 2n² + 4n - 390= 0
⇒ n² + 2n - 195 = 0
⇒ n² + 15n - 13n - 195 = 0
⇒ n( n + 15 ) - 13( n + 15 ) = 0
⇒ ( n + 15 ) ( n - 13 ) = 0
⇒ n = 13 and n ≠ - 15
∴ The two natural consecutive odd numbers be 13 and (13 + 2) = 15 .
∴ the sum of the numbers = 13 + 15 = 28
Quicker Approach:
By mental operation, 13² + 15² = 169 + 225 = 394
∴ Required sum = 13 + 15 = 28