Solve using the quadratic formula.

4v2 − 5v − 1 = 0

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

v = 
 or v = 


We are given the quadratic equation:

\[
4v^2 - 5v - 1 = 0
\]

To solve this using the quadratic formula, we will use the following formula:

\[
v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(4v^2 - 5v - 1 = 0\), the coefficients are:
- \(a = 4\)
- \(b = -5\)
- \(c = -1\)

### Step 1: Calculate the discriminant
The discriminant is given by \(b^2 - 4ac\).

\[
\text{Discriminant} = (-5)^2 - 4(4)(-1) = 25 + 16 = 41
\]

### Step 2: Plug the values into the quadratic formula

\[
v = \frac{-(-5) \pm \sqrt{41}}{2(4)} = \frac{5 \pm \sqrt{41}}{8}
\]

### Step 3: Find the two solutions

Now, we calculate the two possible values for \(v\):

1. \(v_1 = \frac{5 + \sqrt{41}}{8}\)
2. \(v_2 = \frac{5 - \sqrt{41}}{8}\)

Using a calculator to approximate \(\sqrt{41} \approx 6.403\), we get:

1. \(v_1 = \frac{5 + 6.403}{8} = \frac{11.403}{8} \approx 1.43\)
2. \(v_2 = \frac{5 - 6.403}{8} = \frac{-1.403}{8} \approx -0.18\)

### Final Answer:
\[
v_1 \approx 1.43 \quad \text{or} \quad v_2 \approx -0.18
\]

Would you like a more detailed explanation or help with further clarifications? 

Here are 5 questions to expand your understanding:
1. How does the discriminant affect the type of solutions in a quadratic equation?
2. What is the significance of the \(\pm\) symbol in the quadratic formula?
3. Can you derive the quadratic formula from completing the square?
4. What happens if the discriminant is zero in a quadratic equation?
5. How does rounding affect the accuracy of the final answer?

**Tip:** Always check the sign of the discriminant to quickly determine if the quadratic equation has real, distinct, or complex solutions.

Solve using the quadratic formula. 4v^2 − 5v − 1 = 0. Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

Learn how to solve the quadratic equation 4v^2 - 5v - 1 using the quadratic formula. This step-by-step guide includes calculations for the discriminant and the final solutions, suitable for students in Grades 9-10. Solutions are rounded to the nearest hundredth.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation