To solve the quadratic equation \(5x^2 + 14x - 48 = 0\), we can use the **quadratic formula**:

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

Here, the coefficients are:
- \(a = 5\)
- \(b = 14\)
- \(c = -48\)

Let's plug these values into the formula and calculate the roots.

First, compute the discriminant \( \Delta = b^2 - 4ac \):

\[
\Delta = 14^2 - 4(5)(-48)
\]

Now I'll calculate the value of \( \Delta \).The discriminant \( \Delta = 1156 \).

Now, we can calculate the two possible solutions for \(x\) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{1156}}{2a}
\]

Since \( \sqrt{1156} = 34 \), we now compute both roots:

1. \( x_1 = \frac{-b + 34}{2a} \)
2. \( x_2 = \frac{-b - 34}{2a} \)

I'll calculate these values now.The solutions to the quadratic equation \(5x^2 + 14x - 48 = 0\) are:

\[
x_1 = 2.0 \quad \text{and} \quad x_2 = -4.8
\]

Would you like more details or explanations on this process?

Here are 5 related questions for further exploration:
1. What does the discriminant tell us about the nature of the roots?
2. How do you factor a quadratic equation manually?
3. Can you graph the function \( f(x) = 5x^2 + 14x - 48 \) and identify its roots?
4. How is the quadratic formula derived?
5. What happens if the discriminant is negative or zero?

**Tip:** The sign of the discriminant (\( \Delta \)) determines the nature of the roots: two distinct real roots if positive, one real root if zero, and complex roots if negative.

Learn how to solve the quadratic equation 5x^2 + 14x - 48 = 0 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the use of the quadratic formula and discriminant analysis. Suitable for students in Grades 9-10.

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