Find All Complex Solutions 0=3x^2+2x+1

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

Step 1

Move all the expressions to the left side of the equation.

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

$- 3 x^{2} = 2 x + 1$

Subtract $2 x$ from both sides of the equation.

$- 3 x^{2} - 2 x = 1$

Subtract $1$ from both sides of the equation.

$- 3 x^{2} - 2 x - 1 = 0$

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 $a = - 3$ , $b = - 2$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $- 3$ .

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

Multiply $12$ by $- 1$ .

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

Subtract $12$ from $4$ .

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

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

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

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

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

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

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

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

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Multiply $2$ by $- 3$ .

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

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

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

Move the negative in front of the fraction.

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

Find All Complex Solutions 0=3x^2+2x+1

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