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

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

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 $a = 2$ , $b = - 6$ , and $c = 3$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot 2 \cdot 3$ .

Multiply $- 4$ by $2$ .

$x = \frac{6 \pm \sqrt{36 - 8 \cdot 3}}{2 \cdot 2}$

Multiply $- 8$ by $3$ .

$x = \frac{6 \pm \sqrt{36 - 24}}{2 \cdot 2}$

Subtract $24$ from $36$ .

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

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $2$ .

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

Simplify $\frac{6 \pm 2 \sqrt{3}}{4}$ .

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

Step 4

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{3}}{2}, \frac{3 - \sqrt{3}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{3}}{2}, \frac{3 - \sqrt{3}}{2}$

Decimal Form:

$x = 2.36602540 \dots, 0.63397459 \dots$