Solve Using the Quadratic Formula 3x-2x^2=1

$3 x - 2 x^{2} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$3 x - 2 x^{2} - 1 = 0$

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 $a = - 2$ , $b = 3$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot - 2 \cdot - 1$ .

Multiply $- 4$ by $- 2$ .

$x = \frac{- 3 \pm \sqrt{9 + 8 \cdot - 1}}{2 \cdot - 2}$

Multiply $8$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 - 8}}{2 \cdot - 2}$

Subtract $8$ from $9$ .

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

Any root of $1$ is $1$ .

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

Multiply $2$ by $- 2$ .

$x = \frac{- 3 \pm 1}{- 4}$

Simplify $\frac{- 3 \pm 1}{- 4}$ .

$x = \frac{3 \pm 1}{4}$

Step 5

The final answer is the combination of both solutions.

$x = 1, \frac{1}{2}$