The sum of the squares of two consecutive positive integers is 365 . Find the integers.

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The sum of the squares of two consecutive positive integers is 365 . Find the integers.

Let the required two consecutive positive integers be $x$ and $(x+1)$ .

According to the given condition,

$\begin{array}{l} x^{2}+(x+1)^{2}=365 \\ \Rightarrow x^{2}+x^{2}+2 x+1=365 \\ \Rightarrow 2 x^{2}+2 x-364=0 \\ \Rightarrow x^{2}+x-182=0 \\ \Rightarrow x^{2}+14 x-13 x-182=0 \\ \Rightarrow x(x+14)-13(x+14)=0 \\ \Rightarrow(x+14)(x-13)=0 \\ \Rightarrow x+14=0 \text { or } x-13=0 \\ \Rightarrow x=-14 \text { or } x=13 \\ \therefore x=13 \quad \quad(x \text { is a positive integers) } \\ \text { When } x=13 \\ x+1=13+1=14 \end{array}$