To solve the quadratic equation \(x^2 - 5x - 120 = 0\), we can use the **quadratic formula**:

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

For the equation \(x^2 - 5x - 120 = 0\), the coefficients are:

- \(a = 1\)
- \(b = -5\)
- \(c = -120\)

Now, substitute these values into the quadratic formula:

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-120)}}{2(1)}
\]

This simplifies to:

\[
x = \frac{5 \pm \sqrt{25 + 480}}{2}
\]
\[
x = \frac{5 \pm \sqrt{505}}{2}
\]

Since \(\sqrt{505}\) is approximately 22.47, we can further simplify:

\[
x = \frac{5 \pm 22.47}{2}
\]

Thus, the two possible solutions are:

\[
x_1 = \frac{5 + 22.47}{2} = \frac{27.47}{2} = 13.735
\]
\[
x_2 = \frac{5 - 22.47}{2} = \frac{-17.47}{2} = -8.735
\]

So, the solutions are approximately:

\[
x_1 \approx 13.735, \quad x_2 \approx -8.735
\]

Would you like further details or have any questions?

Here are five related questions for deeper exploration:

1. How do you derive the quadratic formula?
2. What is the significance of the discriminant in the quadratic formula?
3. Can this quadratic be solved by factoring?
4. How would you check the solutions for accuracy?
5. How is the quadratic formula applied in real-world scenarios?

**Tip:** Always check if the quadratic equation can be factored before using the quadratic formula. It can save time!

Learn how to solve the quadratic equation x^2 - 5x - 120 = 0 using the quadratic formula. This step-by-step guide explores key algebraic concepts, the discriminant, and provides an accurate solution for students in Grades 9-10.

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