Algebra Examples

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

Step 1

Factor each term.

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Step 1.1

Remove unnecessary parentheses.

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

Step 1.2

Factor.

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Step 1.2.1

Factor out negative.

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

Step 1.2.2

Remove unnecessary parentheses.

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

Step 2

Find the LCD of the terms in the equation.

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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

Remove parentheses.

$1, x^{2} + 1$

Step 2.3

The LCM of one and any expression is the expression.

$x^{2} + 1$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

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

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Step 3.2.2.1

Move $x^{2}$ .

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

Step 3.2.2.2

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

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Step 3.2.2.2.1

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

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

Step 3.2.2.2.2

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

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

Step 3.2.2.3

Add $2$ and $1$ .

$- f x^{3} - f x \cdot 1 = \frac{x}{x^{2} + 1} (x^{2} + 1)$

Step 3.2.3

Multiply $- 1$ by $1$ .

$- f x^{3} - f x = \frac{x}{x^{2} + 1} (x^{2} + 1)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $x^{2} + 1$ .

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Step 3.3.1.1

Cancel the common factor.

$- f x^{3} - f x = \frac{x}{x^{2} + 1} (x^{2} + 1)$

Step 3.3.1.2

Rewrite the expression.

$- f x^{3} - f x = x$

Step 4

Solve the equation.

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Step 4.1

Subtract $x$ from both sides of the equation.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Factor $x$ out of $- f x$ .

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

Step 4.2.3

Factor $x$ out of $- x$ .

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

Step 4.2.4

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

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

Step 4.2.5

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

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

Step 4.3

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

$x = 0$

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

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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Step 4.5.2.1

Move all terms not containing $x$ to the right side of the equation.

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Step 4.5.2.1.1

Add $f$ to both sides of the equation.

$- f x^{2} - 1 = f$

Step 4.5.2.1.2

Add $1$ to both sides of the equation.

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

Step 4.5.2.2

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

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Step 4.5.2.2.1

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

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

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 4.5.2.2.2.2

Cancel the common factor of $f$ .

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Step 4.5.2.2.2.2.1

Cancel the common factor.

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

Step 4.5.2.2.2.2.2

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

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

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Simplify each term.

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Step 4.5.2.2.3.1.1

Cancel the common factor of $f$ .

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Step 4.5.2.2.3.1.1.1

Cancel the common factor.

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

Step 4.5.2.2.3.1.1.2

Rewrite the expression.

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

Step 4.5.2.2.3.1.1.3

Move the negative one from the denominator of $\frac{1}{- 1}$ .

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

Step 4.5.2.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 4.5.2.2.3.1.3

Cancel the common factor of $1$ and $- 1$ .

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Step 4.5.2.2.3.1.3.1

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

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

Step 4.5.2.2.3.1.3.2

Move the negative in front of the fraction.

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

Step 4.5.2.3

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

$x = \pm \sqrt{- 1 - \frac{1}{f}}$

Step 4.5.2.4

Simplify $\pm \sqrt{- 1 - \frac{1}{f}}$ .

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Step 4.5.2.4.1

Factor $- 1$ out of $- 1 - \frac{1}{f}$ .

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Step 4.5.2.4.1.1

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

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

Step 4.5.2.4.1.2

Factor $- 1$ out of $- \frac{1}{f}$ .

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

Step 4.5.2.4.1.3

Factor $- 1$ out of $- 1 (1) - (\frac{1}{f})$ .

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

Step 4.5.2.4.1.4

Rewrite $- 1 (1 + \frac{1}{f})$ as $- (1 + \frac{1}{f})$ .

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

Step 4.5.2.4.2

Write $1$ as a fraction with a common denominator.

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

Step 4.5.2.4.3

Combine the numerators over the common denominator.

$x = \pm \sqrt{- \frac{f + 1}{f}}$

Step 4.5.2.5

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

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Step 4.5.2.5.1

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

$x = \sqrt{- \frac{f + 1}{f}}$

Step 4.5.2.5.2

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