Solve by Factoring 4x^2-8x=3

$4 x^{2} - 8 x = 3$

Step 1

Subtract $3$ from both sides of the equation.

$4 x^{2} - 8 x - 3 = 0$

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 $a = 4$ , $b = - 8$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (4 \cdot - 3)}}{2 \cdot 4}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 4 \cdot - 3}}{2 \cdot 4}$

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

Multiply $- 4$ by $4$ .

$x = \frac{8 \pm \sqrt{64 - 16 \cdot - 3}}{2 \cdot 4}$

Multiply $- 16$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 + 48}}{2 \cdot 4}$

Add $64$ and $48$ .

$x = \frac{8 \pm \sqrt{112}}{2 \cdot 4}$

Rewrite $112$ as $4^{2} \cdot 7$ .

Factor $16$ out of $112$ .

$x = \frac{8 \pm \sqrt{16 (7)}}{2 \cdot 4}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{7}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{8 \pm 4 \sqrt{7}}{8}$

Simplify $\frac{8 \pm 4 \sqrt{7}}{8}$ .

$x = \frac{2 \pm \sqrt{7}}{2}$

Step 5

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

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 4 \cdot - 3}}{2 \cdot 4}$

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

Multiply $- 4$ by $4$ .

$x = \frac{8 \pm \sqrt{64 - 16 \cdot - 3}}{2 \cdot 4}$

Multiply $- 16$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 + 48}}{2 \cdot 4}$

Add $64$ and $48$ .

$x = \frac{8 \pm \sqrt{112}}{2 \cdot 4}$

Rewrite $112$ as $4^{2} \cdot 7$ .

Factor $16$ out of $112$ .

$x = \frac{8 \pm \sqrt{16 (7)}}{2 \cdot 4}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{7}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{8 \pm 4 \sqrt{7}}{8}$

Simplify $\frac{8 \pm 4 \sqrt{7}}{8}$ .

$x = \frac{2 \pm \sqrt{7}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{2 + \sqrt{7}}{2}$

Step 6

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

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 4 \cdot - 3}}{2 \cdot 4}$

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

Multiply $- 4$ by $4$ .

$x = \frac{8 \pm \sqrt{64 - 16 \cdot - 3}}{2 \cdot 4}$

Multiply $- 16$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 + 48}}{2 \cdot 4}$

Add $64$ and $48$ .

$x = \frac{8 \pm \sqrt{112}}{2 \cdot 4}$

Rewrite $112$ as $4^{2} \cdot 7$ .

Factor $16$ out of $112$ .

$x = \frac{8 \pm \sqrt{16 (7)}}{2 \cdot 4}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{7}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{8 \pm 4 \sqrt{7}}{8}$

Simplify $\frac{8 \pm 4 \sqrt{7}}{8}$ .

$x = \frac{2 \pm \sqrt{7}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{2 - \sqrt{7}}{2}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{2 + \sqrt{7}}{2}, \frac{2 - \sqrt{7}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 + \sqrt{7}}{2}, \frac{2 - \sqrt{7}}{2}$

Decimal Form:

$x = 2.32287565 \dots, - 0.32287565 \dots$

Solve by Factoring 4x^2-8x=3

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