Solve Radical B9De0C

1. **State the problem:** Solve the equation $$\sqrt{x-6} - x = -6$$ for $$x$$. 2. **Isolate the square root:** Add $$x$$ to both sides to get: $$\sqrt{x-6} = x - 6$$ 3. **Square both sides:** To eliminate the square root, square both sides: $$\left(\sqrt{x-6}\right)^2 = (x - 6)^2$$ which simplifies to $$x - 6 = (x - 6)^2$$ 4. **Rewrite the equation:** $$x - 6 = (x - 6)^2$$ 5. **Set the equation to zero:** $$0 = (x - 6)^2 - (x - 6)$$ 6. **Factor the right side:** $$0 = (x - 6)\left((x - 6) - 1\right) = (x - 6)(x - 7)$$ 7. **Solve for $$x$$:** Set each factor equal to zero: - $$x - 6 = 0 \implies x = 6$$ - $$x - 7 = 0 \implies x = 7$$ 8. **Check for extraneous solutions:** Substitute $$x=6$$ into the original equation: $$\sqrt{6-6} - 6 = 0 - 6 = -6$$ which matches the right side $$-6$$, so $$x=6$$ is valid. Substitute $$x=7$$: $$\sqrt{7-6} - 7 = \sqrt{1} - 7 = 1 - 7 = -6$$ which matches the right side $$-6$$, so $$x=7$$ is valid. 9. **Final answer:** $$x = 6, 7$$ (in order from least to greatest).