Calculus Examples

$2 x + (\frac{3456}{x}) = 0$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$1, x, 1$

Step 1.2

The LCM of one and any expression is the expression.

$x$

Step 2

Multiply each term in $2 x + \frac{3456}{x} = 0$ by $x$ to eliminate the fractions.

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Step 2.1

Multiply each term in $2 x + \frac{3456}{x} = 0$ by $x$ .

$2 x \cdot x + \frac{3456}{x} x = 0 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Move $x$ .

$2 (x \cdot x) + \frac{3456}{x} x = 0 x$

Step 2.2.1.1.2

Multiply $x$ by $x$ .

$2 x^{2} + \frac{3456}{x} x = 0 x$

Step 2.2.1.2

Cancel the common factor of $x$ .

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Step 2.2.1.2.1

Cancel the common factor.

$2 x^{2} + \frac{3456}{x} x = 0 x$

Step 2.2.1.2.2

Rewrite the expression.

$2 x^{2} + 3456 = 0 x$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $0$ by $x$ .

$2 x^{2} + 3456 = 0$

Step 3

Solve the equation.

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Step 3.1

Subtract $3456$ from both sides of the equation.

$2 x^{2} = - 3456$

Step 3.2

Divide each term in $2 x^{2} = - 3456$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x^{2} = - 3456$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{- 3456}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{- 3456}{2}$

Step 3.2.2.1.2

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

$x^{2} = \frac{- 3456}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 3456$ by $2$ .

$x^{2} = - 1728$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1728}$

Step 3.4

Simplify $\pm \sqrt{- 1728}$ .

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Step 3.4.1

Rewrite $- 1728$ as $- 1 (1728)$ .

$x = \pm \sqrt{- 1 (1728)}$

Step 3.4.2

Rewrite $\sqrt{- 1 (1728)}$ as $\sqrt{- 1} \cdot \sqrt{1728}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{1728}$

Step 3.4.3

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

$x = \pm i \cdot \sqrt{1728}$

Step 3.4.4

Rewrite $1728$ as $24^{2} \cdot 3$ .

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Step 3.4.4.1

Factor $576$ out of $1728$ .

$x = \pm i \cdot \sqrt{576 (3)}$

Step 3.4.4.2

Rewrite $576$ as $24^{2}$ .

$x = \pm i \cdot \sqrt{24^{2} \cdot 3}$

Step 3.4.5

Pull terms out from under the radical.

$x = \pm i \cdot (24 \sqrt{3})$

Step 3.4.6

Move $24$ to the left of $i$ .

$x = \pm 24 i \sqrt{3}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 24 i \sqrt{3}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 24 i \sqrt{3}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 24 i \sqrt{3}, - 24 i \sqrt{3}$