Basic Math Examples

$- 3 n^{2} - 4 n = 3 n^{2} + 2$

Step 1

Move all terms containing $n$ to the left side of the equation.

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Step 1.1

Subtract $3 n^{2}$ from both sides of the equation.

$- 3 n^{2} - 4 n - 3 n^{2} = 2$

Step 1.2

Subtract $3 n^{2}$ from $- 3 n^{2}$ .

$- 6 n^{2} - 4 n = 2$

Step 2

Subtract $2$ from both sides of the equation.

$- 6 n^{2} - 4 n - 2 = 0$

Step 3

Factor $- 2$ out of $- 6 n^{2} - 4 n - 2$ .

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Step 3.1

Factor $- 2$ out of $- 6 n^{2}$ .

$- 2 (3 n^{2}) - 4 n - 2 = 0$

Step 3.2

Factor $- 2$ out of $- 4 n$ .

$- 2 (3 n^{2}) - 2 (2 n) - 2 = 0$

Step 3.3

Factor $- 2$ out of $- 2$ .

$- 2 (3 n^{2}) - 2 (2 n) - 2 \cdot 1 = 0$

Step 3.4

Factor $- 2$ out of $- 2 (3 n^{2}) - 2 (2 n)$ .

$- 2 (3 n^{2} + 2 n) - 2 \cdot 1 = 0$

Step 3.5

Factor $- 2$ out of $- 2 (3 n^{2} + 2 n) - 2 (1)$ .

$- 2 (3 n^{2} + 2 n + 1) = 0$

Step 4

Divide each term in $- 2 (3 n^{2} + 2 n + 1) = 0$ by $- 2$ and simplify.

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Step 4.1

Divide each term in $- 2 (3 n^{2} + 2 n + 1) = 0$ by $- 2$ .

$\frac{- 2 (3 n^{2} + 2 n + 1)}{- 2} = \frac{0}{- 2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 2 (3 n^{2} + 2 n + 1)}{- 2} = \frac{0}{- 2}$

Step 4.2.1.2

Divide $3 n^{2} + 2 n + 1$ by $1$ .

$3 n^{2} + 2 n + 1 = \frac{0}{- 2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 2$ .

$3 n^{2} + 2 n + 1 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 3$ , $b = 2$ , and $c = 1$ into the quadratic formula and solve for $n$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (3 \cdot 1)}}{2 \cdot 3}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$n = \frac{- 2 \pm \sqrt{4 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$

Step 7.1.2

Multiply $- 4 \cdot 3 \cdot 1$ .

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Step 7.1.2.1

Multiply $- 4$ by $3$ .

$n = \frac{- 2 \pm \sqrt{4 - 12 \cdot 1}}{2 \cdot 3}$

Step 7.1.2.2

Multiply $- 12$ by $1$ .

$n = \frac{- 2 \pm \sqrt{4 - 12}}{2 \cdot 3}$

Step 7.1.3

Subtract $12$ from $4$ .

$n = \frac{- 2 \pm \sqrt{- 8}}{2 \cdot 3}$

Step 7.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$n = \frac{- 2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 3}$

Step 7.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$n = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 3}$

Step 7.1.6

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

$n = \frac{- 2 \pm i \cdot \sqrt{8}}{2 \cdot 3}$

Step 7.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.1.7.1

Factor $4$ out of $8$ .

$n = \frac{- 2 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 3}$

Step 7.1.7.2

Rewrite $4$ as $2^{2}$ .

$n = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 3}$

Step 7.1.8

Pull terms out from under the radical.

$n = \frac{- 2 \pm i \cdot (2 \sqrt{2})}{2 \cdot 3}$

Step 7.1.9

Move $2$ to the left of $i$ .

$n = \frac{- 2 \pm 2 i \sqrt{2}}{2 \cdot 3}$

Step 7.2

Multiply $2$ by $3$ .

$n = \frac{- 2 \pm 2 i \sqrt{2}}{6}$

Step 7.3

Simplify $\frac{- 2 \pm 2 i \sqrt{2}}{6}$ .

$n = \frac{- 1 \pm i \sqrt{2}}{3}$

Step 8

The final answer is the combination of both solutions.

$n = - \frac{1 - i \sqrt{2}}{3}, - \frac{1 + i \sqrt{2}}{3}$