Calculus Examples

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

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

Remove parentheses.

$1, x + 1$

Step 1.3

The LCM of one and any expression is the expression.

$x + 1$

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

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

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Step 2.2.2.1

Move $x$ .

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

Step 2.2.2.2

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

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

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

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

Step 2.2.2.3

Add $1$ and $2$ .

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

Step 2.2.3

Multiply $f$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $x + 1$ .

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Step 2.3.1.1

Cancel the common factor.

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

Step 2.3.1.2

Rewrite the expression.

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

Step 3

Solve the equation.

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Step 3.1

Subtract $x$ from both sides of the equation.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Factor $x$ out of $- x$ .

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.3

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

$x = 0$

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

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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Step 3.5.2.1

Use the quadratic formula to find the solutions.

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

Step 3.5.2.2

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

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

Step 3.5.2.3

Simplify the numerator.

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Step 3.5.2.3.1

Factor $f$ out of $f^{2} - 4 \cdot f \cdot - 1$ .

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Step 3.5.2.3.1.1

Factor $f$ out of $f^{2}$ .

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

Step 3.5.2.3.1.2

Factor $f$ out of $- 4 \cdot f \cdot - 1$ .

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

Step 3.5.2.3.1.3

Factor $f$ out of $f \cdot f + f (- 4 \cdot - 1)$ .

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

Step 3.5.2.3.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- f \pm \sqrt{f (f + 4)}}{2 f}$

Step 3.5.2.4

The final answer is the combination of both solutions.