The sum of the squares of two consecutive odd natural numbers is 290 . Find the numbers.

Answer

The required two consecutive odd natural numbers are 11 and 13

Explanation

Let the two consecutive odd natural numbers be x and x+2.
From the given condition.
`x^(2) + ( x+ 2)^(2) = 290`
`:. x^(2) + x^(2) + 4x + 4 = 290 `
`:. 2x^(2) + 4x + 4 - 290 = 0 `
`:. 2x^(2) + 4x - 286 =0 `
`:. x^(2) + 2x - 143 = 0 ` ...( Dividing by 2)
`:. x^(2) + 13x - 11x - 143 = 0 `
` :. x( x+13) - 11( x+ 13) =0 `
`:. ( x+ 13) ( x-11) = 0 `
` :. x + 13s = 0 ` or ` x- 11 = 0 `
`:. x = -13 ` or ` x= 11`
But -13 is not a natural number.
`:. x = -13` is unacceptable.
`:. x = 11 ` and `x + 2 = 11 + 2 = 13 `