1. **State the problem:** Solve the equation $$x^2 + (5+x)^2 - 166 = 2400$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$x^2 + (5+x)^2 - 166 - 2400 = 0$$ which simplifies to $$x^2 + (5+x)^2 - 2566 = 0$$ 3. **Expand the squared term:** $$(5+x)^2 = 25 + 10x + x^2$$ So the equation becomes $$x^2 + 25 + 10x + x^2 - 2566 = 0$$ 4. **Combine like terms:** $$x^2 + x^2 + 10x + 25 - 2566 = 0$$ $$2x^2 + 10x - 2541 = 0$$ 5. **Simplify the quadratic equation:** Divide the entire equation by 2 to simplify: $$\cancel{2}x^2 + \cancel{2}5x - \cancel{2}2541 = 0 \Rightarrow x^2 + 5x - 1270.5 = 0$$ 6. **Use the quadratic formula:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=5$, and $c=-1270.5$. 7. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4(1)(-1270.5) = 25 + 5082 = 5107$$ 8. **Find the roots:** $$x = \frac{-5 \pm \sqrt{5107}}{2}$$ 9. **Approximate the square root:** $$\sqrt{5107} \approx 71.47$$ 10. **Calculate the two solutions:** $$x_1 = \frac{-5 + 71.47}{2} = \frac{66.47}{2} = 33.235$$ $$x_2 = \frac{-5 - 71.47}{2} = \frac{-76.47}{2} = -38.235$$ **Final answer:** $$x \approx 33.235 \text{ or } x \approx -38.235$$