Question: 1. Simplify $\sqrt{81} + 2\sqrt{5} + \sqrt{63} + \sqrt{45}$ as much as possible.
2. For $f(x) = x^2 + 4x - 2$
a) Use the discriminant to determine the nature of the roots.
b) Calculate the exact x-intercepts of $f(x)$. Show your work.
1. Simplify $\sqrt{81} + 2\sqrt{5} + \sqrt{63} + \sqrt{45}$.
2. For $f(x) = x^2 + 4x - 2$:
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### Problem 1: Simplify the expression
1. The problem is to simplify $\sqrt{81} + 2\sqrt{5} + \sqrt{63} + \sqrt{45}$.
2. Recall the rule: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, and simplify perfect squares inside radicals.
3. Simplify each radical:
- $\sqrt{81} = 9$ because $81 = 9^2$.
- $2\sqrt{5}$ stays as is since $5$ is prime.
- $\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}$.
- $\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$.
4. Substitute back:
$$9 + 2\sqrt{5} + 3\sqrt{7} + 3\sqrt{5}$$
5. Combine like terms (radicals with the same radicand):
$$2\sqrt{5} + 3\sqrt{5} = (2 + 3)\sqrt{5} = 5\sqrt{5}$$
6. Final simplified expression:
$$9 + 5\sqrt{5} + 3\sqrt{7}$$
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### Problem 2a: Nature of roots using the discriminant
1. Given $f(x) = x^2 + 4x - 2$, identify coefficients:
$$a = 1, \quad b = 4, \quad c = -2$$
2. The discriminant formula is:
$$\Delta = b^2 - 4ac$$
3. Calculate:
$$\Delta = 4^2 - 4 \times 1 \times (-2) = 16 + 8 = 24$$
4. Since $\Delta > 0$ and not a perfect square, the roots are two distinct real irrational numbers.
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### Problem 2b: Calculate exact x-intercepts
1. Use the quadratic formula:
$$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$
2. Substitute values:
$$x = \frac{-4 \pm \sqrt{24}}{2}$$
3. Simplify $\sqrt{24}$:
$$\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$$
4. Substitute back:
$$x = \frac{-4 \pm 2\sqrt{6}}{2}$$
5. Simplify the fraction by canceling 2:
$$x = \frac{\cancel{2}(-2 \pm \sqrt{6})}{\cancel{2}} = -2 \pm \sqrt{6}$$
6. Final exact x-intercepts:
$$x = -2 + \sqrt{6} \quad \text{and} \quad x = -2 - \sqrt{6}$$