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Bot GPT 3.5 Let the two integers be $x$ and $y$. We know that $|x-y|=15$ and $x^2+y^2=1225$. We are looking for two integers that differ by 15 and have a sum of squares of 1225.
If $x>y$, then $x-y=15$, so $y=x-15$. Substituting that into the second equation $x^2+y^2=1225$, we get $x^2+(x-15)^2=1225$. Expanding and rearranging, we have $2x^2-30x+350=1225$. Subtracting 1225 from both sides, we get $2x^2-30x-875=0$. Factoring a 2, we have $2(x^2-15x-437)=0$. We now need to solve $x^2-15x-437=0$.
The quadratic formula is $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, so we have $x=\frac{15\pm\sqrt{(-15)^2-4(1)(-437)}}{2(1)}$. Simplifying, we have $x=\frac{15\pm\sqrt{15^2+4(437)}}{2}$. Continuing to simplify, we have $x=\frac{15\pm\sqrt{225+1748}}{2}$. We can simplify further to arrive at $x=\frac{15\pm\sqrt{1973}}{2}$. Because we want to find integer solutions, the only possible value of $x$ is $\frac{15+\sqrt{1973}}{2}$.
If $x<y$, then $y-x=15$, so $y=x+15$. Substituting that into the second equation $x^2+y^2=1225$, we get $x^2+(x+15)^2=1225$. Expanding and rearranging, we have $2x^2+30x+325=1225$. Subtracting 1225 from both sides, we get $2x^2+30x-900=0$. Factoring a 2, we have $2(x^2+15x-450)=0$. We now need to solve $x^2+15x-450=0$.
Again, using the quadratic formula, we have $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Therefore, we have $x=\frac{-15\pm\sqrt{(15)^2-4(1)(-450)}}{2(1)}$. Simplifying, we have $x=\frac{-15\pm\sqrt{225+1800}}{2}$. Continuing to simplify, we have $x=\frac{-15\pm\sqrt{2025}}{2}$. We can simplify further to arrive at $x=\frac{-15\pm45}{2}$. The two values are $x=15$ and $x=-30$.
Therefore, the two pairs of integers that satisfy the conditions are $(15, 0)$ and $(0, -15)$.