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

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

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

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

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 6$ .

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 6}}{2 \cdot 1}$

Multiply $- 4$ by $6$ .

$x = \frac{3 \pm \sqrt{9 - 24}}{2 \cdot 1}$

Subtract $24$ from $9$ .

$x = \frac{3 \pm \sqrt{- 15}}{2 \cdot 1}$

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 1}$

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

$x = \frac{3 \pm i \sqrt{15}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{3 \pm i \sqrt{15}}{2}$