To solve the quadratic equation \( n^2 + 48n - 880 = 0 \), we can use the quadratic formula, which is given by:

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

For our equation \( n^2 + 48n - 880 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = 48 \)
- \( c = -880 \)

Now, substitute these values into the quadratic formula:

\[
n = \frac{-48 \pm \sqrt{48^2 - 4 \cdot 1 \cdot (-880)}}{2 \cdot 1}
\]

Calculate the discriminant (\( b^2 - 4ac \)):

\[
b^2 = 48^2 = 2304
\]
\[
4ac = 4 \cdot 1 \cdot (-880) = -3520
\]
\[
b^2 - 4ac = 2304 - (-3520) = 2304 + 3520 = 5824
\]

Now substitute the discriminant back into the formula:

\[
n = \frac{-48 \pm \sqrt{5824}}{2}
\]

Simplify the square root of 5824:

\[
\sqrt{5824} = \sqrt{4 \times 1456} = 2 \sqrt{1456} = 2 \times 4 \sqrt{91} = 8 \sqrt{91}
\]

Thus, we have:

\[
n = \frac{-48 \pm 8\sqrt{91}}{2}
\]

Simplify further:

\[
n = -24 \pm 4\sqrt{91}
\]

This gives us two solutions:

\[
n_1 = -24 + 4\sqrt{91}
\]
\[
n_2 = -24 - 4\sqrt{91}
\]

These are the solutions to the quadratic equation \( n^2 + 48n - 880 = 0 \).

Would you like more details or have any questions?

### Related Questions:
1. How do you derive the quadratic formula?
2. What are some methods other than the quadratic formula to solve a quadratic equation?
3. How can you determine if a quadratic equation has real or complex solutions?
4. Can the discriminant be negative? What does that imply about the roots?
5. How do you factor a quadratic equation if it can be factored?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) first to understand the nature of the roots before solving the quadratic equation.

Explore how to solve the quadratic equation n^2 + 48n - 880 = 0 step-by-step. Learn the quadratic formula method, calculating the discriminant, and deriving the solutions. Suitable for Grades 10-12 students and anyone interested in quadratic equations.

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