To solve the quadratic equation \(25x^2 - 20x - 6 = 0\) using the quadratic formula, we use the formula:

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

where \(a = 25\), \(b = -20\), and \(c = -6\). Let's substitute these values into the formula:

1. **Calculate the discriminant (\(D\)):**

\[
D = b^2 - 4ac = (-20)^2 - 4 \cdot 25 \cdot (-6)
\]

\[
D = 400 + 600 = 1000
\]

2. **Calculate the roots:**

\[
x = \frac{-(-20) \pm \sqrt{1000}}{2 \cdot 25}
\]

\[
x = \frac{20 \pm \sqrt{1000}}{50}
\]

Now, let's simplify \(\sqrt{1000}\):

\[
\sqrt{1000} = \sqrt{100 \times 10} = 10\sqrt{10}
\]

3. **Substitute back and find the two solutions:**

\[
x = \frac{20 \pm 10\sqrt{10}}{50}
\]

\[
x = \frac{20}{50} \pm \frac{10\sqrt{10}}{50}
\]

\[
x = \frac{2}{5} \pm \frac{\sqrt{10}}{5}
\]

Therefore, the solutions are:

\[
x_1 = \frac{2 + \sqrt{10}}{5}, \quad x_2 = \frac{2 - \sqrt{10}}{5}
\]

### Summary:
- \(x_1 = \frac{2 + \sqrt{10}}{5}\)
- \(x_2 = \frac{2 - \sqrt{10}}{5}\)

Would you like more details or have any questions?

### Related Questions:
1. What is the quadratic formula, and how is it derived?
2. How can you determine the nature of the roots using the discriminant?
3. How do you solve quadratic equations by completing the square?
4. What are the possible scenarios for the discriminant value (positive, zero, negative)?
5. How can quadratic equations be applied in real-life problems?

### Tip:
Always check the discriminant (\(b^2 - 4ac\)) first to determine the number and type of solutions a quadratic equation has before solving.

Learn how to solve the quadratic equation 25x² - 20x - 6 step-by-step using the quadratic formula. Explore important concepts like the discriminant and roots, suitable for students in Grades 9-12.

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