To solve the quadratic equation \( 3x^2 - x - 5 = 0 \), we'll use the quadratic formula:

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

where \( a = 3 \), \( b = -1 \), and \( c = -5 \).

### Step 1: Identify coefficients
- \( a = 3 \)
- \( b = -1 \)
- \( c = -5 \)

### Step 2: Apply the quadratic formula

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-5)}}{2(3)}
\]

\[
x = \frac{1 \pm \sqrt{1 + 60}}{6}
\]

\[
x = \frac{1 \pm \sqrt{61}}{6}
\]

### Step 3: Find the two solutions
The solutions are:

\[
x_1 = \frac{1 + \sqrt{61}}{6}
\]

\[
x_2 = \frac{1 - \sqrt{61}}{6}
\]

These are the exact solutions. If you want approximate decimal values:

\[
x_1 \approx 1.567
\]
\[
x_2 \approx -1.234
\]

Would you like more details or have any questions about this solution?

Here are 5 related questions:
1. How do you determine if a quadratic equation has real or complex roots?
2. What does the discriminant (\( b^2 - 4ac \)) tell you about the nature of the roots?
3. Can you factorize the quadratic equation \( 3x^2 - x - 5 = 0 \) instead of using the quadratic formula?
4. What are the steps for completing the square for this quadratic equation?
5. How does the graph of \( 3x^2 - x - 5 = 0 \) look, and where does it intersect the x-axis?

**Tip:** Always check the discriminant (\( b^2 - 4ac \)) before solving; it indicates whether you'll get real or complex roots.

Discover how to solve the quadratic equation 3x^2 - x - 5 = 0 step-by-step using the quadratic formula. This problem demonstrates key algebraic concepts such as discriminant analysis and real roots calculation. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation