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

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

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

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

Reorder terms.

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 2$ , $b = - 4$ , and $c = 3$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

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

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

Multiply $- 4$ by $2$ .

$x = \frac{4 \pm \sqrt{16 - 8 \cdot 3}}{2 \cdot 2}$

Multiply $- 8$ by $3$ .

$x = \frac{4 \pm \sqrt{16 - 24}}{2 \cdot 2}$

Subtract $24$ from $16$ .

$x = \frac{4 \pm \sqrt{- 8}}{2 \cdot 2}$

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 2}$

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

$x = \frac{4 \pm i \cdot \sqrt{8}}{2 \cdot 2}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

Simplify the numerator.

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

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

Multiply $- 4$ by $2$ .

$x = \frac{4 \pm \sqrt{16 - 8 \cdot 3}}{2 \cdot 2}$

Multiply $- 8$ by $3$ .

$x = \frac{4 \pm \sqrt{16 - 24}}{2 \cdot 2}$

Subtract $24$ from $16$ .

$x = \frac{4 \pm \sqrt{- 8}}{2 \cdot 2}$

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 2}$

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

$x = \frac{4 \pm i \cdot \sqrt{8}}{2 \cdot 2}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Multiply $2$ by $2$ .

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

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

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

Change the $\pm$ to $+$ .

$x = \frac{2 + i \sqrt{2}}{2}$

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

Simplify the numerator.

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

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

Multiply $- 4$ by $2$ .

$x = \frac{4 \pm \sqrt{16 - 8 \cdot 3}}{2 \cdot 2}$

Multiply $- 8$ by $3$ .

$x = \frac{4 \pm \sqrt{16 - 24}}{2 \cdot 2}$

Subtract $24$ from $16$ .

$x = \frac{4 \pm \sqrt{- 8}}{2 \cdot 2}$

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 2}$

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

$x = \frac{4 \pm i \cdot \sqrt{8}}{2 \cdot 2}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Multiply $2$ by $2$ .

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

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

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

Change the $\pm$ to $-$ .

$x = \frac{2 - i \sqrt{2}}{2}$

The final answer is the combination of both solutions.

$x = \frac{2 + i \sqrt{2}}{2}, \frac{2 - i \sqrt{2}}{2}$

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

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