Solve Using the Quadratic Formula 1/2n^2+2/3n=-1

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

Move all terms to the left side of the equation and simplify.

Simplify the left side.

Simplify each term.

Combine $\frac{1}{2}$ and $n^{2}$ .

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

Combine $\frac{2}{3}$ and $n$ .

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

Add $1$ to both sides of the equation.

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

Multiply through by the least common denominator $6$ , then simplify.

Apply the distributive property.

$6 (\frac{n^{2}}{2}) + 6 (\frac{2 n}{3}) + 6 \cdot 1 = 0$

Simplify.

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$2 (3) (\frac{n^{2}}{2}) + 6 (\frac{2 n}{3}) + 6 \cdot 1 = 0$

Cancel the common factor.

$2 \cdot (3 (\frac{n^{2}}{2})) + 6 (\frac{2 n}{3}) + 6 \cdot 1 = 0$

Rewrite the expression.

$3 n^{2} + 6 (\frac{2 n}{3}) + 6 \cdot 1 = 0$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$3 n^{2} + 3 (2) (\frac{2 n}{3}) + 6 \cdot 1 = 0$

Cancel the common factor.

$3 n^{2} + 3 \cdot (2 (\frac{2 n}{3})) + 6 \cdot 1 = 0$

Rewrite the expression.

$3 n^{2} + 2 (2 n) + 6 \cdot 1 = 0$

Multiply $2$ by $2$ .

$3 n^{2} + 4 n + 6 \cdot 1 = 0$

Multiply $6$ by $1$ .

$3 n^{2} + 4 n + 6 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 3$ , $b = 4$ , and $c = 6$ into the quadratic formula and solve for $n$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (3 \cdot 6)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$n = \frac{- 4 \pm \sqrt{16 - 4 \cdot 3 \cdot 6}}{2 \cdot 3}$

Multiply $- 4$ by $3$ .

$n = \frac{- 4 \pm \sqrt{16 - 12 \cdot 6}}{2 \cdot 3}$

Multiply $- 12$ by $6$ .

$n = \frac{- 4 \pm \sqrt{16 - 72}}{2 \cdot 3}$

Subtract $72$ from $16$ .

$n = \frac{- 4 \pm \sqrt{- 56}}{2 \cdot 3}$

Rewrite $- 56$ as $- 1 (56)$ .

$n = \frac{- 4 \pm \sqrt{- 1 \cdot 56}}{2 \cdot 3}$

Rewrite $\sqrt{- 1 (56)}$ as $\sqrt{- 1} \cdot \sqrt{56}$ .

$n = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{56}}{2 \cdot 3}$

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

$n = \frac{- 4 \pm i \cdot \sqrt{56}}{2 \cdot 3}$

Rewrite $56$ as $2^{2} \cdot 14$ .

Factor $4$ out of $56$ .

$n = \frac{- 4 \pm i \cdot \sqrt{4 (14)}}{2 \cdot 3}$

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

$n = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 14}}{2 \cdot 3}$

Pull terms out from under the radical.

$n = \frac{- 4 \pm i \cdot (2 \sqrt{14})}{2 \cdot 3}$

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

$n = \frac{- 4 \pm 2 i \sqrt{14}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$n = \frac{- 4 \pm 2 i \sqrt{14}}{6}$

Simplify $\frac{- 4 \pm 2 i \sqrt{14}}{6}$ .

$n = \frac{- 2 \pm i \sqrt{14}}{3}$

Solve Using the Quadratic Formula 1/2n^2+2/3n=-1

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