1. **State the problem:** Solve the equation $x - 6\sqrt{x} = 16$ for $x$. 2. **Rewrite the equation:** Let $y = \sqrt{x}$. Then $x = y^2$. Substitute into the equation: $$y^2 - 6y = 16$$ 3. **Bring all terms to one side:** $$y^2 - 6y - 16 = 0$$ 4. **Solve the quadratic equation:** Use the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1$, $b=-6$, $c=-16$. Calculate the discriminant: $$\Delta = (-6)^2 - 4 \times 1 \times (-16) = 36 + 64 = 100$$ Calculate the roots: $$y = \frac{6 \pm \sqrt{100}}{2} = \frac{6 \pm 10}{2}$$ So, - $y_1 = \frac{6 + 10}{2} = 8$ - $y_2 = \frac{6 - 10}{2} = -2$ 5. **Check for valid solutions:** Since $y = \sqrt{x}$, $y$ must be non-negative. So discard $y = -2$. 6. **Find $x$:** $$x = y^2 = 8^2 = 64$$ 7. **Verify the solution:** Substitute $x=64$ back into the original equation: $$64 - 6\sqrt{64} = 64 - 6 \times 8 = 64 - 48 = 16$$ This is true. **Final answer:** $$x = 64$$