Solve Using the Quadratic Formula 3x^2=4x-9

$3 x^{2} = 4 x - 9$

Step 1

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

Subtract $4 x$ from both sides of the equation.

$3 x^{2} - 4 x = - 9$

Add $9$ to both sides of the equation.

$3 x^{2} - 4 x + 9 = 0$

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 $a = 3$ , $b = - 4$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (3 \cdot 9)}}{2 \cdot 3}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 3 \cdot 9}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 9$ .

Multiply $- 4$ by $3$ .

$x = \frac{4 \pm \sqrt{16 - 12 \cdot 9}}{2 \cdot 3}$

Multiply $- 12$ by $9$ .

$x = \frac{4 \pm \sqrt{16 - 108}}{2 \cdot 3}$

Subtract $108$ from $16$ .

$x = \frac{4 \pm \sqrt{- 92}}{2 \cdot 3}$

Rewrite $- 92$ as $- 1 (92)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 92}}{2 \cdot 3}$

Rewrite $\sqrt{- 1 (92)}$ as $\sqrt{- 1} \cdot \sqrt{92}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{92}}{2 \cdot 3}$

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

$x = \frac{4 \pm i \cdot \sqrt{92}}{2 \cdot 3}$

Rewrite $92$ as $2^{2} \cdot 23$ .

Factor $4$ out of $92$ .

$x = \frac{4 \pm i \cdot \sqrt{4 (23)}}{2 \cdot 3}$

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2} \cdot 23}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{4 \pm i \cdot (2 \sqrt{23})}{2 \cdot 3}$

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

$x = \frac{4 \pm 2 i \sqrt{23}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{4 \pm 2 i \sqrt{23}}{6}$

Simplify $\frac{4 \pm 2 i \sqrt{23}}{6}$ .

$x = \frac{2 \pm i \sqrt{23}}{3}$