Solve Using the Quadratic Formula 5x^2-2x+4=0

$5 x^{2} - 2 x + 4 = 0$

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 $a = 5$ , $b = - 2$ , and $c = 4$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $5$ .

$x = \frac{2 \pm \sqrt{4 - 20 \cdot 4}}{2 \cdot 5}$

Multiply $- 20$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 80}}{2 \cdot 5}$

Subtract $80$ from $4$ .

$x = \frac{2 \pm \sqrt{- 76}}{2 \cdot 5}$

Rewrite $- 76$ as $- 1 (76)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 76}}{2 \cdot 5}$

Rewrite $\sqrt{- 1 (76)}$ as $\sqrt{- 1} \cdot \sqrt{76}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{76}}{2 \cdot 5}$

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

$x = \frac{2 \pm i \cdot \sqrt{76}}{2 \cdot 5}$

Rewrite $76$ as $2^{2} \cdot 19$ .

Factor $4$ out of $76$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (19)}}{2 \cdot 5}$

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 19}}{2 \cdot 5}$

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{19})}{2 \cdot 5}$

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{19}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{2 \pm 2 i \sqrt{19}}{10}$

Simplify $\frac{2 \pm 2 i \sqrt{19}}{10}$ .

$x = \frac{1 \pm i \sqrt{19}}{5}$