1. **Problem statement:** Calculate the values of $x = \sqrt{9 - 4\sqrt{5}}$ and $y = \sqrt{9 + 4\sqrt{5}}$, then evaluate and compare the expression $(9 + \sqrt{5})^2 - (9 - \sqrt{5})^2$. 2. **Recall the difference of squares formula:** $$(a+b)^2 - (a-b)^2 = 4ab$$ This formula will help simplify the expression. 3. **Calculate $x$ and $y$: ** Start by simplifying inside the square roots. For $x$: $$x = \sqrt{9 - 4\sqrt{5}}$$ Try to express $9 - 4\sqrt{5}$ as $(\sqrt{a} - \sqrt{b})^2 = a + b - 2\sqrt{ab}$. Set: $$a + b = 9$$ $$2\sqrt{ab} = 4\sqrt{5} \implies \sqrt{ab} = 2\sqrt{5} \implies ab = 20$$ Solve the system: $$a + b = 9$$ $$ab = 20$$ The quadratic equation for $a$ is: $$a^2 - 9a + 20 = 0$$ Solve: $$a = \frac{9 \pm \sqrt{81 - 80}}{2} = \frac{9 \pm 1}{2}$$ So, $$a = 5 \quad \text{or} \quad a = 4$$ Correspondingly, $$b = 4 \quad \text{or} \quad b = 5$$ Thus, $$9 - 4\sqrt{5} = (\sqrt{5} - \sqrt{4})^2 = (\sqrt{5} - 2)^2$$ Therefore, $$x = \sqrt{(\sqrt{5} - 2)^2} = |\sqrt{5} - 2| = 2 - \sqrt{5}$$ Since $\sqrt{5} \approx 2.236$, $2 - \sqrt{5}$ is negative, so we take the positive value: $$x = \sqrt{5} - 2$$ 4. For $y$: $$y = \sqrt{9 + 4\sqrt{5}}$$ Similarly, express as $(\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab}$. Using the same $a=5$, $b=4$: $$9 + 4\sqrt{5} = (\sqrt{5} + 2)^2$$ So, $$y = \sqrt{(\sqrt{5} + 2)^2} = \sqrt{5} + 2$$ 5. **Evaluate the difference of squares:** $$(9 + \sqrt{5})^2 - (9 - \sqrt{5})^2 = 4 \times 9 \times \sqrt{5} = 36\sqrt{5}$$ 6. **Compare $x$ and $y$:** Since $x = \sqrt{5} - 2 \approx 0.236$ and $y = \sqrt{5} + 2 \approx 4.236$, clearly $y > x$. **Final answers:** $$x = \sqrt{5} - 2$$ $$y = \sqrt{5} + 2$$ $$(9 + \sqrt{5})^2 - (9 - \sqrt{5})^2 = 36\sqrt{5}$$ Therefore, $y$ is greater than $x$ when comparing the speeds.