Calculus Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2

Combine $- 2$ and $\frac{1}{x^{2}}$ .

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

Step 1.3

Move the negative in front of the fraction.

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

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

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

$1, x^{2}, 1$

Step 2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.2.1.1.1

Move $x^{2}$ .

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

Step 3.2.1.1.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 3.2.1.1.2.1

Raise $x$ to the power of $1$ .

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

Step 3.2.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.1.1.3

Add $2$ and $1$ .

$2 x^{3} - \frac{2}{x^{2}} x^{2} = 0 x^{2}$

Step 3.2.1.2

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 3.2.1.2.1

Move the leading negative in $- \frac{2}{x^{2}}$ into the numerator.

$2 x^{3} + \frac{- 2}{x^{2}} x^{2} = 0 x^{2}$

Step 3.2.1.2.2

Cancel the common factor.

$2 x^{3} + \frac{- 2}{x^{2}} x^{2} = 0 x^{2}$

Step 3.2.1.2.3

Rewrite the expression.

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

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Multiply $0$ by $x^{2}$ .

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

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Add $2$ to both sides of the equation.

$2 x^{3} = 2$

Step 4.2

Subtract $2$ from both sides of the equation.

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

Step 4.3

Factor the left side of the equation.

Tap for more steps...

Step 4.3.1

Factor $2$ out of $2 x^{3} - 2$ .

Tap for more steps...

Step 4.3.1.1

Factor $2$ out of $2 x^{3}$ .

$2 (x^{3}) - 2 = 0$

Step 4.3.1.2

Factor $2$ out of $- 2$ .

$2 (x^{3}) + 2 (- 1) = 0$

Step 4.3.1.3

Factor $2$ out of $2 (x^{3}) + 2 (- 1)$ .

$2 (x^{3} - 1) = 0$

Step 4.3.2

Rewrite $1$ as $1^{3}$ .

$2 (x^{3} - 1^{3}) = 0$

Step 4.3.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 4.3.4

Factor.

Tap for more steps...

Step 4.3.4.1

Simplify.

Tap for more steps...

Step 4.3.4.1.1

Multiply $x$ by $1$ .

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

Step 4.3.4.1.2

One to any power is one.

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

Step 4.3.4.2

Remove unnecessary parentheses.

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

Step 4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

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

Step 4.5

Set $x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.6

Set $x^{2} + x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.6.1

Set $x^{2} + x + 1$ equal to $0$ .

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

Step 4.6.2

Solve $x^{2} + x + 1 = 0$ for $x$ .

Tap for more steps...

Step 4.6.2.1

Use the quadratic formula to find the solutions.

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

Step 4.6.2.2

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 4.6.2.3

Simplify.

Tap for more steps...

Step 4.6.2.3.1

Simplify the numerator.

Tap for more steps...

Step 4.6.2.3.1.1

One to any power is one.

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

Step 4.6.2.3.1.2

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

Tap for more steps...

Step 4.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 4.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 4.6.2.3.1.3

Subtract $4$ from $1$ .

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

Step 4.6.2.3.1.4

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

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

Step 4.6.2.3.1.5

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

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

Step 4.6.2.3.1.6

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

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

Step 4.6.2.3.2

Multiply $2$ by $1$ .

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

Step 4.6.2.4

The final answer is the combination of both solutions.

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

Step 4.7

The final solution is all the values that make $2 (x - 1) (x^{2} + x + 1) = 0$ true.

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