1. **State the problem:** Solve the equation $\sqrt{4n} + 12 = n$ for $n$. 2. **Rewrite the square root:** Note that $\sqrt{4n} = \sqrt{4} \cdot \sqrt{n} = 2\sqrt{n}$. 3. **Substitute and isolate the square root term:** $$2\sqrt{n} + 12 = n$$ 4. **Isolate the square root:** $$2\sqrt{n} = n - 12$$ 5. **Divide both sides by 2:** $$\sqrt{n} = \frac{n - 12}{2}$$ 6. **Square both sides to eliminate the square root:** $$n = \left(\frac{n - 12}{2}\right)^2 = \frac{(n - 12)^2}{4}$$ 7. **Multiply both sides by 4 to clear the denominator:** $$4n = (n - 12)^2$$ 8. **Expand the right side:** $$4n = n^2 - 24n + 144$$ 9. **Bring all terms to one side:** $$0 = n^2 - 24n + 144 - 4n$$ $$0 = n^2 - 28n + 144$$ 10. **Solve the quadratic equation:** Use the quadratic formula $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=-28$, $c=144$. Calculate the discriminant: $$\Delta = (-28)^2 - 4 \cdot 1 \cdot 144 = 784 - 576 = 208$$ Calculate the roots: $$n = \frac{28 \pm \sqrt{208}}{2} = \frac{28 \pm 4\sqrt{13}}{2} = 14 \pm 2\sqrt{13}$$ 11. **Check for extraneous solutions:** Recall from step 4 that $2\sqrt{n} = n - 12$, so $n - 12 \geq 0 \Rightarrow n \geq 12$. Evaluate approximate values: - $14 + 2\sqrt{13} \approx 14 + 7.21 = 21.21$ (valid) - $14 - 2\sqrt{13} \approx 14 - 7.21 = 6.79$ (invalid since less than 12) 12. **Final answer:** $$\boxed{n = 14 + 2\sqrt{13}}$$