To solve the quadratic equation \(10x^2 - 11x - 6 = 0\), we can use the quadratic formula:

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

Here, \(a = 10\), \(b = -11\), and \(c = -6\).

1. Calculate the discriminant:
   \[
   \text{Discriminant} = b^2 - 4ac = (-11)^2 - 4(10)(-6) = 121 + 240 = 361
   \]

2. Find the square root of the discriminant:
   \[
   \sqrt{361} = 19
   \]

3. Apply the quadratic formula:
   \[
   x = \frac{-(-11) \pm 19}{2(10)} = \frac{11 \pm 19}{20}
   \]

4. Solve for the two possible values of \(x\):
   \[
   x_1 = \frac{11 + 19}{20} = \frac{30}{20} = \frac{3}{2} = 1.5
   \]
   \[
   x_2 = \frac{11 - 19}{20} = \frac{-8}{20} = -0.4
   \]

So, the solutions are:

\[
x = 1.5 \quad \text{or} \quad x = -0.4
\]

Would you like more details or have any questions?

Here are some related questions to explore further:
1. How do you derive the quadratic formula?
2. What is the significance of the discriminant in a quadratic equation?
3. How would the solutions change if the discriminant was negative?
4. Can you solve the quadratic equation by factoring instead of using the quadratic formula?
5. What are the real-world applications of solving quadratic equations?

**Tip:** Always check your solutions by substituting them back into the original equation to verify correctness.

Explore the step-by-step solution to the quadratic equation 10x^2 - 11x - 6 = 0 using the quadratic formula. This problem highlights key algebraic concepts like the discriminant and provides a clear explanation suitable for students in Grades 9-10.

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