To solve the quadratic equation \( 6x^2 + 7x - 3 = 0 \), we can use the quadratic formula:

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

Given:
- \( a = 6 \)
- \( b = 7 \)
- \( c = -3 \)

### Step 1: Calculate the Discriminant

\[
\Delta = b^2 - 4ac = 7^2 - 4 \times 6 \times (-3) = 49 + 72 = 121
\]

### Step 2: Apply the Quadratic Formula

\[
x = \frac{-7 \pm \sqrt{121}}{2 \times 6} = \frac{-7 \pm 11}{12}
\]

### Step 3: Simplify the Results

1. **For the positive square root:**

\[
x_1 = \frac{-7 + 11}{12} = \frac{4}{12} = \frac{1}{3}
\]

2. **For the negative square root:**

\[
x_2 = \frac{-7 - 11}{12} = \frac{-18}{12} = -\frac{3}{2}
\]

### Final Answer

The solutions are:

\[
x = \frac{1}{3} \quad \text{and} \quad x = -\frac{3}{2}
\]

Would you like further details or have any questions?

### Related Questions

1. How do you find the roots of a quadratic equation when the discriminant is negative?
2. How does the sign of the discriminant affect the nature of the roots?
3. Can we factor \( 6x^2 + 7x - 3 \) directly without using the quadratic formula?
4. How would you use completing the square to solve this quadratic equation?
5. What are some real-life applications of solving quadratic equations?

### Tip

Always check the discriminant (\( b^2 - 4ac \)) first when using the quadratic formula. It tells you whether the equation has real or complex roots.

Discover how to solve the quadratic equation 6x^2 + 7x - 3 = 0 using the quadratic formula. This step-by-step guide explains the discriminant, real root calculations, and covers algebraic concepts ideal for Grades 9-10 students.

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