Calculus Examples

$2 x - \frac{128}{x^{2}}$

Step 1

Set $2 x - \frac{128}{x^{2}}$ equal to $0$ .

$2 x - \frac{128}{x^{2}} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Find the LCD of the terms in the equation.

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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.

$1, x^{2}, 1$

Step 2.1.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 2.2

Multiply each term in $2 x - \frac{128}{x^{2}} = 0$ by $x^{2}$ to eliminate the fractions.

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Step 2.2.1

Multiply each term in $2 x - \frac{128}{x^{2}} = 0$ by $x^{2}$ .

$2 x \cdot x^{2} - \frac{128}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

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Step 2.2.2.1.1.1

Move $x^{2}$ .

$2 (x^{2} x) - \frac{128}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.2.1.1.2

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

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Step 2.2.2.1.1.2.1

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

$2 (x^{2} x^{1}) - \frac{128}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.2.1.1.2.2

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

$2 x^{2 + 1} - \frac{128}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.2.1.1.3

Add $2$ and $1$ .

$2 x^{3} - \frac{128}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.2.1.2

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

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Step 2.2.2.1.2.1

Move the leading negative in $- \frac{128}{x^{2}}$ into the numerator.

$2 x^{3} + \frac{- 128}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.2.1.2.2

Cancel the common factor.

$2 x^{3} + \frac{- 128}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.2.1.2.3

Rewrite the expression.

$2 x^{3} - 128 = 0 x^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

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

$2 x^{3} - 128 = 0$

Step 2.3

Solve the equation.

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Step 2.3.1

Add $128$ to both sides of the equation.

$2 x^{3} = 128$

Step 2.3.2

Subtract $128$ from both sides of the equation.

$2 x^{3} - 128 = 0$

Step 2.3.3

Factor the left side of the equation.

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Step 2.3.3.1

Factor $2$ out of $2 x^{3} - 128$ .

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Step 2.3.3.1.1

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

$2 (x^{3}) - 128 = 0$

Step 2.3.3.1.2

Factor $2$ out of $- 128$ .

$2 x^{3} + 2 \cdot - 64 = 0$

Step 2.3.3.1.3

Factor $2$ out of $2 x^{3} + 2 \cdot - 64$ .

$2 (x^{3} - 64) = 0$

Step 2.3.3.2

Rewrite $64$ as $4^{3}$ .

$2 (x^{3} - 4^{3}) = 0$

Step 2.3.3.3

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 $b = 4$ .

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

Step 2.3.3.4

Factor.

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Step 2.3.3.4.1

Simplify.

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Step 2.3.3.4.1.1

Move $4$ to the left of $x$ .

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

Step 2.3.3.4.1.2

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

$2 ((x - 4) (x^{2} + 4 x + 16)) = 0$

Step 2.3.3.4.2

Remove unnecessary parentheses.

$2 (x - 4) (x^{2} + 4 x + 16) = 0$

Step 2.3.4

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

$x - 4 = 0$

$x^{2} + 4 x + 16 = 0$

Step 2.3.5

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.3.5.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.3.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.3.6

Set $x^{2} + 4 x + 16$ equal to $0$ and solve for $x$ .

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Step 2.3.6.1

Set $x^{2} + 4 x + 16$ equal to $0$ .

$x^{2} + 4 x + 16 = 0$

Step 2.3.6.2

Solve $x^{2} + 4 x + 16 = 0$ for $x$ .

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Step 2.3.6.2.1

Use the quadratic formula to find the solutions.

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

Step 2.3.6.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Step 2.3.6.2.3

Simplify.

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Step 2.3.6.2.3.1

Simplify the numerator.

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Step 2.3.6.2.3.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 2.3.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 2.3.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Step 2.3.6.2.3.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 2.3.6.2.3.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 2.3.6.2.3.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 2.3.6.2.3.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 2.3.6.2.3.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 2.3.6.2.3.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.3.6.2.3.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 2.3.6.2.3.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 2.3.6.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 2.3.6.2.3.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 2.3.6.2.3.2

Multiply $2$ by $1$ .

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

Step 2.3.6.2.3.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

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

Step 2.3.6.2.4

The final answer is the combination of both solutions.

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

Step 2.3.7

The final solution is all the values that make $2 (x - 4) (x^{2} + 4 x + 16) = 0$ true.

$x = 4, - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Step 3