Question

Use the quadratic formula to solve. Express your answer in simplest form.
10, n, squared, plus, 21, n, plus, 9, equals, 0
10n
2
+21n+9=
0

Answer 1

To solve the quadratic equation \(10n^2 + 21n + 9 = 0\) using the quadratic formula, we start with the standard form:

\[
ax^2 + bx + c = 0
\]

where:

- \(a = 10\)
- \(b = 21\)
- \(c = 9\)

The quadratic formula is given by:

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

Now, we will calculate the discriminant (\(b^2 - 4ac\)):

\[
b^2 = 21^2 = 441
\]
\[
4ac = 4 \times 10 \times 9 = 360
\]
\[
b^2 - 4ac = 441 - 360 = 81
\]

Now that we have the discriminant, we can substitute back into the quadratic formula:

\[
n = \frac{-21 \pm \sqrt{81}}{2 \cdot 10}
\]

Since \(\sqrt{81} = 9\), we continue:

\[
n = \frac{-21 \pm 9}{20}
\]

Now we solve for the two possible values of \(n\):

1. **Using the positive root:**

\[
n = \frac{-21 + 9}{20} = \frac{-12}{20} = \frac{-3}{5}
\]

2. **Using the negative root:**

\[
n = \frac{-21 - 9}{20} = \frac{-30}{20} = \frac{-3}{2}
\]

Thus, the solutions to the equation \(10n^2 + 21n + 9 = 0\) are:

\[
n = -\frac{3}{5} \quad \text{and} \quad n = -\frac{3}{2}
\]

These are expressed in simplest form as required.