1. **State the problem:** Solve the equation $n^2 = 4(4n + 9)$. 2. **Write the equation and expand:** Start with the given equation: $$n^2 = 4(4n + 9)$$ Expand the right side: $$n^2 = 16n + 36$$ 3. **Bring all terms to one side to set the equation to zero:** $$n^2 - 16n - 36 = 0$$ 4. **Use the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by: $$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=-16$, and $c=-36$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-16)^2 - 4(1)(-36) = 256 + 144 = 400$$ 6. **Find the square root of the discriminant:** $$\sqrt{400} = 20$$ 7. **Calculate the two solutions:** $$n = \frac{-(-16) \pm 20}{2(1)} = \frac{16 \pm 20}{2}$$ 8. **Evaluate each solution:** - For the plus sign: $$n = \frac{16 + 20}{2} = \frac{36}{2} = 18$$ - For the minus sign: $$n = \frac{16 - 20}{2} = \frac{\cancel{16} - 20}{\cancel{2}} = \frac{-4}{2} = -2$$ 9. **Final answer:** The solutions to the equation are: $$n = 18 \text{ or } n = -2$$