Find All Complex Solutions 5x^2-5x+2=0

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 5$ , $b = - 5$ , and $c = 2$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 5 \cdot 2}}{2 \cdot 5}$

Multiply $- 4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 20 \cdot 2}}{2 \cdot 5}$

Multiply $- 20$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 40}}{2 \cdot 5}$

Subtract $40$ from $25$ .

$x = \frac{5 \pm \sqrt{- 15}}{2 \cdot 5}$

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

$x = \frac{5 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 5}$

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

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 5}$

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

$x = \frac{5 \pm i \sqrt{15}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{5 \pm i \sqrt{15}}{10}$