Trigonometry Examples

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

Step 1

Rewrite using the commutative property of multiplication.

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

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.

$1, 2 x^{2} + 1$

Step 2.2

Remove parentheses.

$1, 2 x^{2} + 1$

Step 2.3

The LCM of one and any expression is the expression.

$2 x^{2} + 1$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

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

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Step 3.2.2.1

Move $x^{2}$ .

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

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 $1$ .

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

Step 3.2.2.2.2

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

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

Step 3.2.2.3

Add $2$ and $1$ .

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

Step 3.2.3

Multiply $2$ by $1$ .

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

Step 3.2.4

Multiply $2$ by $2$ .

$4 f x^{3} + 2 f x = \frac{2 x}{2 x^{2} + 1} (2 x^{2} + 1)$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Cancel the common factor.

$4 f x^{3} + 2 f x = \frac{2 x}{2 x^{2} + 1} (2 x^{2} + 1)$

Step 3.3.1.2

Rewrite the expression.

$4 f x^{3} + 2 f x = 2 x$

Step 4

Solve the equation.

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Step 4.1

Subtract $2 x$ from both sides of the equation.

$4 f x^{3} + 2 f x - 2 x = 0$

Step 4.2

Factor $2 x$ out of $4 f x^{3} + 2 f x - 2 x$ .

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Step 4.2.1

Factor $2 x$ out of $4 f x^{3}$ .

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

Step 4.2.2

Factor $2 x$ out of $2 f x$ .

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

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

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

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 $0$ .

$x = 0$

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

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

Subtract $f$ from both sides of the equation.

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

Step 4.5.2.1.2

Add $1$ to both sides of the equation.

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

Step 4.5.2.2

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

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Step 4.5.2.2.1

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

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

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.5.2.2.2.1.1

Cancel the common factor.

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

Step 4.5.2.2.2.1.2

Rewrite the expression.

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

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.

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

Step 4.5.2.2.2.2.2

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

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

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.

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

Step 4.5.2.2.3.1.1.2

Rewrite the expression.

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

Step 4.5.2.2.3.1.2

Move the negative in front of the fraction.

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

Step 4.5.2.3

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

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

Step 4.5.2.4

Simplify $\pm \sqrt{- \frac{1}{2} + \frac{1}{2 f}}$ .

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Step 4.5.2.4.1

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{f}{f}$ .

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

Step 4.5.2.4.2

Multiply $\frac{1}{2}$ by $\frac{f}{f}$ .

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

Step 4.5.2.4.3

Combine the numerators over the common denominator.

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

Step 4.5.2.4.4

Rewrite $\sqrt{\frac{- f + 1}{2 f}}$ as $\frac{\sqrt{- f + 1}}{\sqrt{2 f}}$ .

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

Step 4.5.2.4.5

Multiply $\frac{\sqrt{- f + 1}}{\sqrt{2 f}}$ by $\frac{\sqrt{2 f}}{\sqrt{2 f}}$ .

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

Step 4.5.2.4.6

Combine and simplify the denominator.

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Step 4.5.2.4.6.1

Multiply $\frac{\sqrt{- f + 1}}{\sqrt{2 f}}$ by $\frac{\sqrt{2 f}}{\sqrt{2 f}}$ .

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

Step 4.5.2.4.6.2

Raise $\sqrt{2 f}$ to the power of $1$ .

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

Step 4.5.2.4.6.3

Raise $\sqrt{2 f}$ to the power of $1$ .

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

Step 4.5.2.4.6.4

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

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

Step 4.5.2.4.6.5

Add $1$ and $1$ .

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

Step 4.5.2.4.6.6

Rewrite ${\sqrt{2 f}}^{2}$ as $2 f$ .

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Step 4.5.2.4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 f}$ as ${(2 f)}^{\frac{1}{2}}$ .

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

Step 4.5.2.4.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.5.2.4.6.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.5.2.4.6.6.4

Cancel the common factor of $2$ .

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Step 4.5.2.4.6.6.4.1

Cancel the common factor.

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

Step 4.5.2.4.6.6.4.2

Rewrite the expression.

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

Step 4.5.2.4.6.6.5

Simplify.

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

Step 4.5.2.4.7

Combine using the product rule for radicals.

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

Step 4.5.2.4.8

Reorder factors in $\pm \frac{\sqrt{(- f + 1) \cdot 2 f}}{2 f}$ .

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

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.