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,

x^{2} + (x + 2)^{2} = 514

x^{2} + x^{2} + 4x + 4 = 514 using (a + b)^{2} = a^{2} + 2ab + b^{2}

2x^{2} + 4x – 510 = 0

x^{2} + 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

x^{2} + 2x – 255 = 0

x^{2} + 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

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