Solve Using the Quadratic Formula 7x^2=2x-1

$7 x^{2} = 2 x - 1$

Step 1

Move all the expressions to the left side of the equation.

Subtract $2 x$ from both sides of the equation.

$7 x^{2} - 2 x = - 1$

Add $1$ to both sides of the equation.

$7 x^{2} - 2 x + 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 7$ , $b = - 2$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (7 \cdot 1)}}{2 \cdot 7}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 7 \cdot 1}}{2 \cdot 7}$

Multiply $- 4 \cdot 7 \cdot 1$ .

Multiply $- 4$ by $7$ .

$x = \frac{2 \pm \sqrt{4 - 28 \cdot 1}}{2 \cdot 7}$

Multiply $- 28$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 28}}{2 \cdot 7}$

Subtract $28$ from $4$ .

$x = \frac{2 \pm \sqrt{- 24}}{2 \cdot 7}$

Rewrite $- 24$ as $- 1 (24)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 24}}{2 \cdot 7}$

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot 7}$

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

$x = \frac{2 \pm i \cdot \sqrt{24}}{2 \cdot 7}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (6)}}{2 \cdot 7}$

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot 7}$

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{6})}{2 \cdot 7}$

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

$x = \frac{2 \pm 2 i \sqrt{6}}{2 \cdot 7}$

Multiply $2$ by $7$ .

$x = \frac{2 \pm 2 i \sqrt{6}}{14}$

Simplify $\frac{2 \pm 2 i \sqrt{6}}{14}$ .

$x = \frac{1 \pm i \sqrt{6}}{7}$