Finite Math Examples

$9 n^{2} + 81 = - 18 n$

Step 1

Add $18 n$ to both sides of the equation.

$9 n^{2} + 81 + 18 n = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 9$ , $b = 18$ , and $c = 81$ into the quadratic formula and solve for $n$ .

$\frac{- 18 \pm \sqrt{18^{2} - 4 \cdot (9 \cdot 81)}}{2 \cdot 9}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $18$ to the power of $2$ .

$n = \frac{- 18 \pm \sqrt{324 - 4 \cdot 9 \cdot 81}}{2 \cdot 9}$

Step 4.1.2

Multiply $- 4 \cdot 9 \cdot 81$ .

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Step 4.1.2.1

Multiply $- 4$ by $9$ .

$n = \frac{- 18 \pm \sqrt{324 - 36 \cdot 81}}{2 \cdot 9}$

Step 4.1.2.2

Multiply $- 36$ by $81$ .

$n = \frac{- 18 \pm \sqrt{324 - 2916}}{2 \cdot 9}$

Step 4.1.3

Subtract $2916$ from $324$ .

$n = \frac{- 18 \pm \sqrt{- 2592}}{2 \cdot 9}$

Step 4.1.4

Rewrite $- 2592$ as $- 1 (2592)$ .

$n = \frac{- 18 \pm \sqrt{- 1 \cdot 2592}}{2 \cdot 9}$

Step 4.1.5

Rewrite $\sqrt{- 1 (2592)}$ as $\sqrt{- 1} \cdot \sqrt{2592}$ .

$n = \frac{- 18 \pm \sqrt{- 1} \cdot \sqrt{2592}}{2 \cdot 9}$

Step 4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$n = \frac{- 18 \pm i \cdot \sqrt{2592}}{2 \cdot 9}$

Step 4.1.7

Rewrite $2592$ as $36^{2} \cdot 2$ .

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Step 4.1.7.1

Factor $1296$ out of $2592$ .

$n = \frac{- 18 \pm i \cdot \sqrt{1296 (2)}}{2 \cdot 9}$

Step 4.1.7.2

Rewrite $1296$ as $36^{2}$ .

$n = \frac{- 18 \pm i \cdot \sqrt{36^{2} \cdot 2}}{2 \cdot 9}$

Step 4.1.8

Pull terms out from under the radical.

$n = \frac{- 18 \pm i \cdot (36 \sqrt{2})}{2 \cdot 9}$

Step 4.1.9

Move $36$ to the left of $i$ .

$n = \frac{- 18 \pm 36 i \sqrt{2}}{2 \cdot 9}$

Step 4.2

Multiply $2$ by $9$ .

$n = \frac{- 18 \pm 36 i \sqrt{2}}{18}$

Step 4.3

Simplify $\frac{- 18 \pm 36 i \sqrt{2}}{18}$ .

$n = - 1 \pm 2 i \sqrt{2}$