Calculus Examples

$f (x) = 2 x + \frac{2}{x - 1}$

Step 1

Set $2 x + \frac{2}{x - 1}$ equal to $0$ .

$2 x + \frac{2}{x - 1} = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x - 1, 1$

Step 2.1.2

Remove parentheses.

$1, x - 1, 1$

Step 2.1.3

The LCM of one and any expression is the expression.

$x - 1$

Step 2.2

Multiply each term in $2 x + \frac{2}{x - 1} = 0$ by $x - 1$ to eliminate the fractions.

Tap for more steps...

Step 2.2.1

Multiply each term in $2 x + \frac{2}{x - 1} = 0$ by $x - 1$ .

$2 x (x - 1) + \frac{2}{x - 1} (x - 1) = 0 (x - 1)$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.2.1.1

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 1 + \frac{2}{x - 1} (x - 1) = 0 (x - 1)$

Step 2.2.2.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.2.2.1.2.1

Move $x$ .

$2 (x \cdot x) + 2 x \cdot - 1 + \frac{2}{x - 1} (x - 1) = 0 (x - 1)$

Step 2.2.2.1.2.2

Multiply $x$ by $x$ .

$2 x^{2} + 2 x \cdot - 1 + \frac{2}{x - 1} (x - 1) = 0 (x - 1)$

Step 2.2.2.1.3

Multiply $- 1$ by $2$ .

$2 x^{2} - 2 x + \frac{2}{x - 1} (x - 1) = 0 (x - 1)$

Step 2.2.2.1.4

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 2.2.2.1.4.1

Cancel the common factor.

$2 x^{2} - 2 x + \frac{2}{x - 1} (x - 1) = 0 (x - 1)$

Step 2.2.2.1.4.2

Rewrite the expression.

$2 x^{2} - 2 x + 2 = 0 (x - 1)$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Multiply $0$ by $x - 1$ .

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

Step 2.3

Solve the equation.

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

Factor $2$ out of $2$ .

$2 (x^{2}) + 2 (- x) + 2 (1) = 0$

Step 2.3.1.4

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

$2 (x^{2} - x) + 2 (1) = 0$

Step 2.3.1.5

Factor $2$ out of $2 (x^{2} - x) + 2 (1)$ .

$2 (x^{2} - x + 1) = 0$

Step 2.3.2

Divide each term in $2 (x^{2} - x + 1) = 0$ by $2$ and simplify.

Tap for more steps...

Step 2.3.2.1

Divide each term in $2 (x^{2} - x + 1) = 0$ by $2$ .

$\frac{2 (x^{2} - x + 1)}{2} = \frac{0}{2}$

Step 2.3.2.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} - x + 1)}{2} = \frac{0}{2}$

Step 2.3.2.2.1.2

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

$x^{2} - x + 1 = \frac{0}{2}$

Step 2.3.2.3

Simplify the right side.

Tap for more steps...

Step 2.3.2.3.1

Divide $0$ by $2$ .

$x^{2} - x + 1 = 0$

Step 2.3.3

Use the quadratic formula to find the solutions.

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

Tap for more steps...

Step 2.3.5.1

Simplify the numerator.

Tap for more steps...

Step 2.3.5.1.1

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

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

Step 2.3.5.1.2

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

Tap for more steps...

Step 2.3.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.3.5.1.2.2

Multiply $- 4$ by $1$ .

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

Step 2.3.5.1.3

Subtract $4$ from $1$ .

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

Step 2.3.5.1.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 2.3.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 2.3.5.1.6

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 2.3.5.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 2.3.6

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 3