1. **Problem:** Solve the equation $\sqrt{x-8} - \sqrt{2x-2} + 3 = 0$. 2. **Step 1:** Isolate one square root term: $$\sqrt{x-8} = \sqrt{2x-2} - 3$$ 3. **Step 2:** Square both sides to eliminate the square root on the left: $$x - 8 = (\sqrt{2x-2} - 3)^2$$ 4. **Step 3:** Expand the right side: $$x - 8 = 2x - 2 - 6\sqrt{2x-2} + 9$$ 5. **Step 4:** Rearrange terms to isolate the square root: $$-x - 15 = -6 \sqrt{2x-2}$$ 6. **Step 5:** Divide both sides by $-6$: $$\frac{x+15}{6} = \sqrt{2x-2}$$ 7. **Step 6:** Square both sides again to remove the square root: $$\left(\frac{x+15}{6}\right)^2 = 2x - 2$$ 8. **Step 7:** Simplify and solve the quadratic equation: $$\frac{(x+15)^2}{36} = 2x - 2$$ Multiply both sides by 36: $$ (x+15)^2 = 72x - 72 $$ Expand left side: $$ x^2 + 30x + 225 = 72x - 72 $$ Bring all terms to one side: $$ x^2 + 30x + 225 - 72x + 72 = 0 $$ $$ x^2 - 42x + 297 = 0 $$ 9. **Step 8:** Use quadratic formula: $$ x = \frac{42 \pm \sqrt{(-42)^2 - 4 \times 1 \times 297}}{2} = \frac{42 \pm \sqrt{1764 - 1188}}{2} = \frac{42 \pm \sqrt{576}}{2} $$ $$ x = \frac{42 \pm 24}{2} $$ 10. **Step 9:** Calculate roots: $$ x = \frac{42 + 24}{2} = 33, \quad x = \frac{42 - 24}{2} = 9 $$ 11. **Step 10:** Check solutions in original equation: - For $x=33$: $$ \sqrt{33-8} - \sqrt{2(33)-2} + 3 = \sqrt{25} - \sqrt{64} + 3 = 5 - 8 + 3 = 0 $$ (valid) - For $x=9$: $$ \sqrt{9-8} - \sqrt{18-2} + 3 = \sqrt{1} - \sqrt{16} + 3 = 1 - 4 + 3 = 0 $$ (valid) **Final answer:** $x = 9$ or $x = 33$.