Solve Using the Quadratic Formula 6x^2-2x+1=0

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

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 $a = 6$ , $b = - 2$ , and $c = 1$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $6$ .

$x = \frac{2 \pm \sqrt{4 - 24 \cdot 1}}{2 \cdot 6}$

Multiply $- 24$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 24}}{2 \cdot 6}$

Subtract $24$ from $4$ .

$x = \frac{2 \pm \sqrt{- 20}}{2 \cdot 6}$

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 20}}{2 \cdot 6}$

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{20}}{2 \cdot 6}$

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

$x = \frac{2 \pm i \cdot \sqrt{20}}{2 \cdot 6}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 6}$

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 6}$

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{5})}{2 \cdot 6}$

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

$x = \frac{2 \pm 2 i \sqrt{5}}{2 \cdot 6}$

Multiply $2$ by $6$ .

$x = \frac{2 \pm 2 i \sqrt{5}}{12}$

Simplify $\frac{2 \pm 2 i \sqrt{5}}{12}$ .

$x = \frac{1 \pm i \sqrt{5}}{6}$