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. Start by simplifying each square root where possible:
$$\sqrt{81} = 9$$
$$\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}$$
$$\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$$
2. Substitute these back into the expression:
$$9 + 2\sqrt{5} + 3\sqrt{7} + 3\sqrt{5}$$
3. Combine like terms (terms with $\sqrt{5}$):
$$2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$$
4. Final simplified expression:
$$9 + 5\sqrt{5} + 3\sqrt{7}$$
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### Problem 2a: Use the discriminant to determine the nature of the roots of $f(x) = x^2 + 4x - 2$
1. Recall the quadratic formula discriminant:
$$\Delta = b^2 - 4ac$$
where $a=1$, $b=4$, and $c=-2$.
2. Calculate the discriminant:
$$\Delta = 4^2 - 4 \times 1 \times (-2) = 16 + 8 = 24$$
3. Since $\Delta > 0$, the quadratic has two distinct real roots.
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### Problem 2b: Calculate the exact x-intercepts of $f(x)$
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:
$$x = -2 + \sqrt{6} \quad \text{and} \quad x = -2 - \sqrt{6}$$