To solve the quadratic equation \( 6y^2 - 3y - 1 = 0 \), we can use the quadratic formula, which is given by:

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

Here, the coefficients are:
- \( a = 6 \)
- \( b = -3 \)
- \( c = -1 \)

Substituting these into the quadratic formula:

\[
y = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(6)(-1)}}{2(6)}
\]

Simplifying step by step:

1. Calculate the discriminant:

\[
b^2 - 4ac = (-3)^2 - 4(6)(-1) = 9 + 24 = 33
\]

2. Take the square root of the discriminant:

\[
\sqrt{33} \approx 5.744
\]

3. Plug it back into the quadratic formula:

\[
y = \frac{3 \pm 5.744}{12}
\]

So, the two possible solutions for \( y \) are:

\[
y_1 = \frac{3 + 5.744}{12} \approx \frac{8.744}{12} \approx 0.729
\]
\[
y_2 = \frac{3 - 5.744}{12} \approx \frac{-2.744}{12} \approx -0.229
\]

Thus, the solutions are:

\[
y_1 \approx 0.729, \quad y_2 \approx -0.229
\]

Would you like more details on any step or have any questions?

### Related Questions:
1. What is the significance of the discriminant in the quadratic formula?
2. How can you verify the solutions of a quadratic equation?
3. What are the implications if the discriminant is negative?
4. How do you solve a quadratic equation using the method of completing the square?
5. What are the graphical interpretations of the solutions to a quadratic equation?

**Tip:** Always check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

This page explains how to solve the quadratic equation 6y^2 - 3y - 1 = 0 using the quadratic formula. The solution includes a step-by-step breakdown of calculating the discriminant and finding the two possible values for y. Suitable for students in Grades 9-12 studying algebra.

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