To solve the quadratic equation \(6x^2 - 7x - 20 = 0\), we can use the quadratic formula:

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

Here, \(a = 6\), \(b = -7\), and \(c = -20\).

Let's calculate the discriminant first:

\[
\text{Discriminant} = b^2 - 4ac = (-7)^2 - 4(6)(-20) = 49 + 480 = 529
\]

Now, apply the quadratic formula:

\[
x = \frac{-(-7) \pm \sqrt{529}}{2(6)} = \frac{7 \pm 23}{12}
\]

This gives us two solutions:

1. \(x_1 = \frac{7 + 23}{12} = \frac{30}{12} = \frac{5}{2} = 2.5\)
2. \(x_2 = \frac{7 - 23}{12} = \frac{-16}{12} = -\frac{4}{3}\)

So, the solutions are:

\[
x = \frac{5}{2} \quad \text{or} \quad x = -\frac{4}{3}
\]

Would you like more details on any part of this solution?

Here are some related questions:

1. How can you determine the nature of roots using the discriminant?
2. What are the steps to factorize a quadratic equation manually?
3. How does completing the square work for solving quadratic equations?
4. Can the quadratic formula be used for any quadratic equation?
5. What are the geometric interpretations of the solutions on the graph?

**Tip:** When solving quadratic equations, always check the discriminant first to determine the nature of the roots.

Learn how to solve the quadratic equation 6x² - 7x - 20 = 0 with a detailed step-by-step solution using the quadratic formula. Understand the discriminant and its role in determining the nature of roots. Suitable for high school students.

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