Solve by Factoring 2x^2+3x+6=4x

$2 x^{2} + 3 x + 6 = 4 x$

Step 1

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

$2 x^{2} + 3 x + 6 - 4 x = 0$

Step 2

Subtract $4 x$ from $3 x$ .

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

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

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

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

Step 5

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

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

Multiply $- 4 \cdot 2 \cdot 6$ .

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot 6}}{2 \cdot 2}$

Multiply $- 8$ by $6$ .

$x = \frac{1 \pm \sqrt{1 - 48}}{2 \cdot 2}$

Subtract $48$ from $1$ .

$x = \frac{1 \pm \sqrt{- 47}}{2 \cdot 2}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 2}$

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

$x = \frac{1 \pm i \sqrt{47}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{1 \pm i \sqrt{47}}{4}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

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

Multiply $- 4 \cdot 2 \cdot 6$ .

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot 6}}{2 \cdot 2}$

Multiply $- 8$ by $6$ .

$x = \frac{1 \pm \sqrt{1 - 48}}{2 \cdot 2}$

Subtract $48$ from $1$ .

$x = \frac{1 \pm \sqrt{- 47}}{2 \cdot 2}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 2}$

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

$x = \frac{1 \pm i \sqrt{47}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{1 \pm i \sqrt{47}}{4}$

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{47}}{4}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

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

Multiply $- 4 \cdot 2 \cdot 6$ .

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot 6}}{2 \cdot 2}$

Multiply $- 8$ by $6$ .

$x = \frac{1 \pm \sqrt{1 - 48}}{2 \cdot 2}$

Subtract $48$ from $1$ .

$x = \frac{1 \pm \sqrt{- 47}}{2 \cdot 2}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 2}$

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

$x = \frac{1 \pm i \sqrt{47}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{1 \pm i \sqrt{47}}{4}$

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{47}}{4}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{47}}{4}, \frac{1 - i \sqrt{47}}{4}$

Solve by Factoring 2x^2+3x+6=4x

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