Algebra Examples

$2 x^{2} - 38 = 4 x$

Step 1

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

$2 x^{2} - 38 - 4 x = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $2$ out of $2 x^{2} - 38 - 4 x$ .

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Step 2.1.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) - 38 - 4 x = 0$

Step 2.1.2

Factor $2$ out of $- 38$ .

$2 x^{2} + 2 \cdot - 19 - 4 x = 0$

Step 2.1.3

Factor $2$ out of $- 4 x$ .

$2 x^{2} + 2 \cdot - 19 + 2 (- 2 x) = 0$

Step 2.1.4

Factor $2$ out of $2 x^{2} + 2 \cdot - 19$ .

$2 (x^{2} - 19) + 2 (- 2 x) = 0$

Step 2.1.5

Factor $2$ out of $2 (x^{2} - 19) + 2 (- 2 x)$ .

$2 (x^{2} - 19 - 2 x) = 0$

Step 2.2

Reorder terms.

$2 (x^{2} - 2 x - 19) = 0$

Step 3

Divide each term in $2 (x^{2} - 2 x - 19) = 0$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 (x^{2} - 2 x - 19) = 0$ by $2$ .

$\frac{2 (x^{2} - 2 x - 19)}{2} = \frac{0}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} - 2 x - 19)}{2} = \frac{0}{2}$

Step 3.2.1.2

Divide $x^{2} - 2 x - 19$ by $1$ .

$x^{2} - 2 x - 19 = \frac{0}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $2$ .

$x^{2} - 2 x - 19 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 19$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 19)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 19}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 19$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 19}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 19$ .

$x = \frac{2 \pm \sqrt{4 + 76}}{2 \cdot 1}$

Step 6.1.3

Add $4$ and $76$ .

$x = \frac{2 \pm \sqrt{80}}{2 \cdot 1}$

Step 6.1.4

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 6.1.4.1

Factor $16$ out of $80$ .

$x = \frac{2 \pm \sqrt{16 (5)}}{2 \cdot 1}$

Step 6.1.4.2

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

$x = \frac{2 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 4 \sqrt{5}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 4 \sqrt{5}}{2}$

Step 6.3

Simplify $\frac{2 \pm 4 \sqrt{5}}{2}$ .

$x = 1 \pm 2 \sqrt{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 1 \pm 2 \sqrt{5}$

Decimal Form:

$x = 5.47213595 \dots, - 3.47213595 \dots$

Step 8