Find All Complex Solutions 3x^2=3x-5

$3 x^{2} = 3 x - 5$

Step 1

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

Subtract $3 x$ from both sides of the equation.

$3 x^{2} - 3 x = - 5$

Add $5$ to both sides of the equation.

$3 x^{2} - 3 x + 5 = 0$

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 $a = 3$ , $b = - 3$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (3 \cdot 5)}}{2 \cdot 3}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{3 \pm \sqrt{9 - 12 \cdot 5}}{2 \cdot 3}$

Multiply $- 12$ by $5$ .

$x = \frac{3 \pm \sqrt{9 - 60}}{2 \cdot 3}$

Subtract $60$ from $9$ .

$x = \frac{3 \pm \sqrt{- 51}}{2 \cdot 3}$

Rewrite $- 51$ as $- 1 (51)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 51}}{2 \cdot 3}$

Rewrite $\sqrt{- 1 (51)}$ as $\sqrt{- 1} \cdot \sqrt{51}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{51}}{2 \cdot 3}$

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

$x = \frac{3 \pm i \sqrt{51}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{3 \pm i \sqrt{51}}{6}$