Question

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

The sum of the squares of two consecutive odd positive integers is 394. Find them.

Answer 1

Let one of the number be x then the other number be x + 2.

Then according to question,

\[x^2 + \left( x + 2 \right)^2 = 394\]

\[ \Rightarrow x^2 + x^2 + 4x + 4 = 394\]

\[ \Rightarrow 2 x^2 + 4x - 390 = 0\]

\[ \Rightarrow x^2 + 2x - 195 = 0\]

\[ \Rightarrow x^2 + 15x - 13x - 195 = 0\]

\[ \Rightarrow x(x + 15) - 13(x + 15) = 0\]

\[ \Rightarrow (x - 13)(x + 15) = 0\]

\[ \Rightarrow x - 13 = 0 \text { or } x + 15 = 0\]

\[ \Rightarrow x = 13 \text { or } x = - 15\]

Since, x being an odd number,

Therefore, x = 13.

Then another number will be \[x + 2 = 13 + 2 = 15\]

Thus, the two consecutive odd numbers are 13 and 15.