Question

Find two consecutive odd positive integers, sum of whose squares is 514

Answer 1

Solutions :-

Given : Sum of squares = 514

Let one of the odd positive integer be x 
So, other odd positive integer is x+2

Now,
The sum of squares = x² +(x+2)²
                               = x² + x² + 4x +4
                               = 2x² + 4x + 4

⇒ 2x² +4x + 4 = 514
⇒ 2x² +4x = 514 - 4 = 510
⇒ 2x² + 4x - 510 = 0 
⇒ 2(x² + 2x - 255) = 0 
⇒ x² + 2x - 255 = 0
⇒ x² + 17x - 15x -143 = 0 
⇒ x (x+17) - 15 (x+17) = 0 
⇒ (x + 17) (x - 15) = 0
⇒ x = 15 or -17
We always take positive value of x


Answer :
one of the odd positive integer = x = 15
other odd positive integer = x + 2 = 15 + 2 = 17