Question: User: $\sqrt{9 + 4\sqrt{5}} = $
1. **State the problem:** Simplify the expression $$\sqrt{9 + 4\sqrt{5}}$$.
2. **Recall the formula:** Expressions of the form $$\sqrt{a + b}$$ where $$b$$ involves a square root can sometimes be simplified as $$\sqrt{x} + \sqrt{y}$$ such that:
$$\left(\sqrt{x} + \sqrt{y}\right)^2 = x + y + 2\sqrt{xy} = a + b$$.
3. **Set up equations:** We want to find $$x$$ and $$y$$ such that:
$$x + y = 9$$
$$2\sqrt{xy} = 4\sqrt{5}$$
4. **Simplify the second equation:**
$$2\sqrt{xy} = 4\sqrt{5} \implies \sqrt{xy} = 2\sqrt{5} \implies xy = 4 \times 5 = 20$$
5. **Solve the system:**
From $$x + y = 9$$ and $$xy = 20$$, consider $$x$$ and $$y$$ as roots of the quadratic:
$$t^2 - 9t + 20 = 0$$
6. **Find roots:**
$$t = \frac{9 \pm \sqrt{81 - 80}}{2} = \frac{9 \pm 1}{2}$$
So,
$$t_1 = 5, \quad t_2 = 4$$
7. **Check:**
$$\sqrt{5} + \sqrt{4} = \sqrt{5} + 2$$
Square it:
$$\left(\sqrt{5} + 2\right)^2 = 5 + 4 + 2 \times 2 \times \sqrt{5} = 9 + 4\sqrt{5}$$
8. **Final answer:**
$$\sqrt{9 + 4\sqrt{5}} = 2 + \sqrt{5}$$