The sum of the squares to two consecutive positive odd numbers is 514 . Find the numbers.

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The sum of the squares to two consecutive positive odd numbers is 514 . Find the numbers.

Let the two consecutive positive odd numbers be $x$ and $(x+2)$ .
According to the given condition,

$\begin{array}{l} x^{2}+(x+2)^{2}=514 \\ \Rightarrow x^{2}+x^{2}+4 x+4=514 \\ \Rightarrow 2 x^{2}+4 x-510=0 \\ \Rightarrow x^{2}+2 x-255=0 \\ \Rightarrow x^{2}+17 x-15 x-255=0 \\ \Rightarrow x(x+17)-15(x+17)=0 \\ \Rightarrow(x+17)(x-15)=0 \\ \Rightarrow x+17=0 \text { or } x-15=0 \\ \Rightarrow x=-17 \text { or } x=15 \\ \therefore x=15 \quad(x \text { is a positive odd number) } \\ \text { When } x=15 \\ x+2=15+2=17 \end{array}$