Find All Complex Solutions 3x^2-6x+1=0

$3 x^{2} - 6 x + 1 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 3$ , $b = - 6$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (3 \cdot 1)}}{2 \cdot 3}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$

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

Multiply $- 4$ by $3$ .

$x = \frac{6 \pm \sqrt{36 - 12 \cdot 1}}{2 \cdot 3}$

Multiply $- 12$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 12}}{2 \cdot 3}$

Subtract $12$ from $36$ .

$x = \frac{6 \pm \sqrt{24}}{2 \cdot 3}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$x = \frac{6 \pm \sqrt{4 (6)}}{2 \cdot 3}$

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{6}}{2 \cdot 3}$

Multiply $2$ by $3$ .

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

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

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

Step 4

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{6}}{3}, \frac{3 - \sqrt{6}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{6}}{3}, \frac{3 - \sqrt{6}}{3}$

Decimal Form:

$x = 1.81649658 \dots, 0.18350341 \dots$