1. **State the problem:** Solve the equation $$\sqrt{x^2 - 6x + 36} = 10$$. 2. **Formula and rules:** To solve an equation with a square root, first isolate the square root expression (already isolated here), then square both sides to eliminate the square root. 3. **Square both sides:** $$\left(\sqrt{x^2 - 6x + 36}\right)^2 = 10^2$$ $$x^2 - 6x + 36 = 100$$ 4. **Simplify the quadratic equation:** $$x^2 - 6x + 36 - 100 = 0$$ $$x^2 - 6x - 64 = 0$$ 5. **Factor the quadratic:** We look for two numbers that multiply to $$-64$$ and add to $$-6$$. These are $$-16$$ and $$10$$. $$x^2 - 6x - 64 = (x - 16)(x + 10) = 0$$ 6. **Solve for $$x$$:** Set each factor equal to zero: $$x - 16 = 0 \Rightarrow x = 16$$ $$x + 10 = 0 \Rightarrow x = -10$$ 7. **Check for extraneous solutions:** Substitute $$x = 16$$: $$\sqrt{16^2 - 6(16) + 36} = \sqrt{256 - 96 + 36} = \sqrt{196} = 14 \neq 10$$ (reject) Substitute $$x = -10$$: $$\sqrt{(-10)^2 - 6(-10) + 36} = \sqrt{100 + 60 + 36} = \sqrt{196} = 14 \neq 10$$ (reject) Both solutions give $$14$$ under the square root, not $$10$$, so no solution satisfies the original equation. **Final answer:** No real solution exists for the equation $$\sqrt{x^2 - 6x + 36} = 10$$.