1. **State the problem:** Find two consecutive odd numbers such that the sum of their squares is 220. 2. **Define variables:** Let the first odd number be $x$. Since the numbers are consecutive odd numbers, the next odd number is $x + 2$. 3. **Write the equation:** The sum of their squares is given by $$x^2 + (x+2)^2 = 220$$ 4. **Expand and simplify:** $$x^2 + (x^2 + 4x + 4) = 220$$ $$2x^2 + 4x + 4 = 220$$ 5. **Bring all terms to one side:** $$2x^2 + 4x + 4 - 220 = 0$$ $$2x^2 + 4x - 216 = 0$$ 6. **Divide the entire equation by 2 to simplify:** $$x^2 + 2x - 108 = 0$$ 7. **Solve the quadratic equation using the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=2$, and $c=-108$. Calculate the discriminant: $$\Delta = 2^2 - 4 \times 1 \times (-108) = 4 + 432 = 436$$ Calculate the roots: $$x = \frac{-2 \pm \sqrt{436}}{2} = \frac{-2 \pm 2\sqrt{109}}{2} = -1 \pm \sqrt{109}$$ 8. **Approximate the roots:** $$\sqrt{109} \approx 10.44$$ So, $$x_1 = -1 + 10.44 = 9.44$$ $$x_2 = -1 - 10.44 = -11.44$$ 9. **Since $x$ must be an odd integer, check the closest odd integers:** - For $x = 9$, the next odd number is $11$. - For $x = -11$, the next odd number is $-9$. 10. **Verify the sums:** - $9^2 + 11^2 = 81 + 121 = 202$ (not 220) - $-11^2 + (-9)^2 = 121 + 81 = 202$ (not 220) 11. **Check $x=7$ and $x=13$ as possible odd numbers near the roots:** - $7^2 + 9^2 = 49 + 81 = 130$ (too low) - $11^2 + 13^2 = 121 + 169 = 290$ (too high) 12. **Re-examine the problem:** The quadratic solution suggests non-integer roots, but the problem states consecutive odd numbers, which must be integers. 13. **Try to find integer solutions by testing odd numbers:** - $5^2 + 7^2 = 25 + 49 = 74$ - $7^2 + 9^2 = 49 + 81 = 130$ - $9^2 + 11^2 = 81 + 121 = 202$ - $13^2 + 15^2 = 169 + 225 = 394$ No pair sums to 220. 14. **Try negative odd numbers:** - $-15^2 + (-13)^2 = 225 + 169 = 394$ - $-13^2 + (-11)^2 = 169 + 121 = 290$ - $-11^2 + (-9)^2 = 121 + 81 = 202$ No pair sums to 220. 15. **Conclusion:** There are no integer consecutive odd numbers whose squares sum to 220. **Final answer:** No such consecutive odd integers exist.