The sum of the squares of two consecutive odd numbers is 394. Find the numbers. (CBSE 2014)

given: The sum of the squares of two consecutive odd numbers is 394.


To find: the numbers.

Solution:

Let the consecutive odd number be ‘a’ and a + 2


According to given condition,

 a² + (a + 2)² = 394


Use the formula (x+y)²=x²+y²+2xy in (a + 2)²


Here x=a and y=2,


⇒ a² + a² + 4 + 4a=394

⇒ 2a² + 4a +4 – 394 = 0


⇒ 2a² + 4a – 390 = 0


Take 2 common out of the above equation,

⇒ a² + 2a – 195 = 0


Factorise by splitting the middle term.

⇒ a² + 15a – 13a – 195 = 0

⇒ a(a + 15) – 13(a + 15) = 0

⇒ (a – 13)(a + 15) = 0

Thus, a = 13, - 15


When a=13 then a+2=15


And when a = -15 then a+2 = -13

So Consecutive odd numbers are 13, 15 and -15,-13.