The sum of the squares of two consecutive positive odd numbers is 514. Find the numbers.
Let the two consecutive positive odd numbers be x and x + 2
According to given condition,
x2 + (x + 2)2 = 514
x2 + x2 + 4x + 4 = 514 using (a + b)2 = a2 + 2ab + b2
2x2 + 4x – 510 = 0
x2 + 2x – 255 = 0
Using the splitting middle term – the middle term of the general equation 
is divided in two such values that:
Product = a.c
For the given equation a = 1 b = 2 c = – 255
= 1. – 255 = – 255
And either of their sum or difference = b
= 2
Thus the two terms are 17 and – 15
Difference = 17 – 15 = 2
Product = 17. – 15 = – 255
x2 + 2x – 255 = 0
x2 + 17x – 15x – 255 = 0
x(x + 17) – 15(x + 17) = 0
(x + 17) (x – 15) = 0
(x + 17) = 0 or (x – 15) = 0
x = – 17 or x = 15
x = 15 (x is positive odd number)
x + 2 = 15 + 2 = 17
Thus the two consecutive positive odd numbers are 15 and 17