Compute for \( f(x)=\frac{1}{x} \) Homework Help \( \begin{array}{lll}\text { a. } f\left(\frac{1}{2}\right) & \text { b. } f\left(\frac{1}{10}\right) & \text { c. } f(0.01) \\ \text { Solve each of the following quadratic equations. If you need help, refer to the Math Notes box for this lesson. Homework Help } \\ \begin{array}{ll}\text { a. } x^{2}-8 x+15=0 & \text { b. } 2 x^{2}-5 x-6=0\end{array}\end{array} \)

Sure, I'd be happy to help you with your homework! Let's tackle each part step by step. --- ## Part 1: Evaluating the Function \( f(x) = \frac{1}{x} \) ### a. Compute \( f\left(\frac{1}{2}\right) \) \[ f\left(\frac{1}{2}\right) = \frac{1}{\frac{1}{2}} = 2 \] ### b. Compute \( f\left(\frac{1}{10}\right) \) \[ f\left(\frac{1}{10}\right) = \frac{1}{\frac{1}{10}} = 10 \] ### c. Compute \( f(0.01) \) \[ f(0.01) = \frac{1}{0.01} = 100 \] --- ## Part 2: Solving Quadratic Equations ### a. Solve \( x^{2} - 8x + 15 = 0 \) To solve the quadratic equation \( x^{2} - 8x + 15 = 0 \), we can factor it. 1. **Find two numbers that multiply to 15** and **add up to -8**. The numbers are **-3** and **-5** because: \[ (-3) \times (-5) = 15 \quad \text{and} \quad (-3) + (-5) = -8 \] 2. **Factor the quadratic:** \[ x^{2} - 8x + 15 = (x - 3)(x - 5) = 0 \] 3. **Set each factor equal to zero and solve for \( x \):** \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \] **Solutions:** \( x = 3 \) and \( x = 5 \) --- ### b. Solve \( 2x^{2} - 5x - 6 = 0 \) This quadratic equation doesn't factor neatly, so we'll use the **quadratic formula**: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] Where \( a = 2 \), \( b = -5 \), and \( c = -6 \). 1. **Calculate the discriminant (\( \Delta \)):** \[ \Delta = b^{2} - 4ac = (-5)^{2} - 4(2)(-6) = 25 + 48 = 73 \] 2. **Apply the quadratic formula:** \[ x = \frac{-(-5) \pm \sqrt{73}}{2 \times 2} = \frac{5 \pm \sqrt{73}}{4} \] 3. **Simplify (if necessary):** The solutions can be left in the exact form: \[ x = \frac{5 + \sqrt{73}}{4} \quad \text{and} \quad x = \frac{5 - \sqrt{73}}{4} \] Alternatively, you can approximate the solutions using a calculator: \[ \sqrt{73} \approx 8.544 \] \[ x \approx \frac{5 + 8.544}{4} = \frac{13.544}{4} \approx 3.386 \] \[ x \approx \frac{5 - 8.544}{4} = \frac{-3.544}{4} \approx -0.886 \] **Solutions:** - Exact form: \( x = \frac{5 \pm \sqrt{73}}{4} \) - Approximate values: \[ x \approx 3.386 \quad \text{and} \quad x \approx -0.886 \] --- If you have any more questions or need further clarification, feel free to ask!