Question

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

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.