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

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

UpStudy AI · Accepted Answer

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!
a. $$ f\left(\frac{1}{2}\right) = 2 $$ b. $$ f\left(\frac{1}{10}\right) = 10 $$ c. $$ f(0.01) = 100 $$ --- **Solving Quadratic Equations:** a. $$ x^{2} - 8x + 15 = 0 $$ - Solutions: $$ x = 3 $$ and $$ x = 5 $$ b. $$ 2x^{2} - 5x - 6 = 0 $$ - Solutions: $$ x = \frac{5 + \sqrt{73}}{4} $$ and $$ x = \frac{5 - \sqrt{73}}{4} $$ (approximately $$ x \approx 3.386 $$ and $$ x \approx -0.886 $$)

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