1. **State the problem:** Solve the equation $$\sqrt{x-8} - \sqrt{2x-2} + 3 = 0$$ for $x$. 2. **Isolate one of the square roots:** Move terms to isolate one radical: $$\sqrt{x-8} = \sqrt{2x-2} - 3$$ 3. **Square both sides:** To eliminate the square roots, square both sides: $$\left(\sqrt{x-8}\right)^2 = \left(\sqrt{2x-2} - 3\right)^2$$ $$x - 8 = (\sqrt{2x-2})^2 - 2 \cdot 3 \cdot \sqrt{2x-2} + 3^2$$ $$x - 8 = 2x - 2 - 6\sqrt{2x-2} + 9$$ 4. **Simplify the right side:** $$x - 8 = 2x + 7 - 6\sqrt{2x-2}$$ 5. **Rearrange to isolate the radical term:** $$x - 8 - 2x - 7 = -6\sqrt{2x-2}$$ $$-x - 15 = -6\sqrt{2x-2}$$ 6. **Divide both sides by -6:** $$\frac{-x - 15}{-6} = \sqrt{2x-2}$$ $$\cancel{\frac{-x - 15}{-6}} = \sqrt{2x-2}$$ 7. **Square both sides again to eliminate the square root:** $$\left(\frac{x + 15}{6}\right)^2 = 2x - 2$$ $$\frac{(x + 15)^2}{36} = 2x - 2$$ 8. **Multiply both sides by 36 to clear the denominator:** $$ (x + 15)^2 = 36(2x - 2) $$ $$ x^2 + 30x + 225 = 72x - 72 $$ 9. **Bring all terms to one side:** $$ x^2 + 30x + 225 - 72x + 72 = 0 $$ $$ x^2 - 42x + 297 = 0 $$ 10. **Solve the quadratic equation:** Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=-42$, $c=297$. $$x = \frac{42 \pm \sqrt{(-42)^2 - 4 \cdot 1 \cdot 297}}{2}$$ $$x = \frac{42 \pm \sqrt{1764 - 1188}}{2}$$ $$x = \frac{42 \pm \sqrt{576}}{2}$$ $$x = \frac{42 \pm 24}{2}$$ 11. **Calculate the two possible solutions:** $$x_1 = \frac{42 + 24}{2} = \frac{66}{2} = 33$$ $$x_2 = \frac{42 - 24}{2} = \frac{18}{2} = 9$$ 12. **Check for extraneous solutions:** - For $x=33$: $$\sqrt{33 - 8} - \sqrt{2 \cdot 33 - 2} + 3 = \sqrt{25} - \sqrt{64} + 3 = 5 - 8 + 3 = 0$$ (valid) - For $x=9$: $$\sqrt{9 - 8} - \sqrt{2 \cdot 9 - 2} + 3 = \sqrt{1} - \sqrt{16} + 3 = 1 - 4 + 3 = 0$$ (valid) **Final answer:** $$x = 9 \text{ or } x = 33$$