1. **State the problem:** Solve the equation $$\sqrt{x^2 - 4x + 13} + \sqrt{x^2 - 4x + 8} = 5$$ for $x$. 2. **Rewrite the expressions inside the square roots:** Complete the square for each quadratic: $$x^2 - 4x + 13 = (x^2 - 4x + 4) + 9 = (x-2)^2 + 9$$ $$x^2 - 4x + 8 = (x^2 - 4x + 4) + 4 = (x-2)^2 + 4$$ 3. **Substitute $y = x-2$ to simplify:** The equation becomes: $$\sqrt{y^2 + 9} + \sqrt{y^2 + 4} = 5$$ 4. **Isolate one square root:** $$\sqrt{y^2 + 9} = 5 - \sqrt{y^2 + 4}$$ 5. **Square both sides:** $$y^2 + 9 = (5 - \sqrt{y^2 + 4})^2 = 25 - 10\sqrt{y^2 + 4} + y^2 + 4$$ Simplify: $$y^2 + 9 = y^2 + 29 - 10\sqrt{y^2 + 4}$$ 6. **Subtract $y^2$ and 9 from both sides:** $$0 = 20 - 10\sqrt{y^2 + 4}$$ 7. **Divide both sides by 10:** $$0 = 2 - \sqrt{y^2 + 4}$$ 8. **Isolate the square root:** $$\sqrt{y^2 + 4} = 2$$ 9. **Square both sides again:** $$y^2 + 4 = 4$$ 10. **Solve for $y^2$:** $$y^2 = 0$$ 11. **Find $y$:** $$y = 0$$ 12. **Recall substitution $y = x - 2$:** $$x - 2 = 0 \implies x = 2$$ 13. **Check the solution in the original equation:** $$\sqrt{2^2 - 4(2) + 13} + \sqrt{2^2 - 4(2) + 8} = \sqrt{4 - 8 + 13} + \sqrt{4 - 8 + 8} = \sqrt{9} + \sqrt{4} = 3 + 2 = 5$$ The solution satisfies the equation. **Final answer:** $$x = 2$$