Algebra Examples

$\frac{\frac{x + 1}{x + 4} - \frac{1}{x}}{x + 1}$

Step 1

Multiply the numerator and denominator of the fraction by $(x + 4) x$ .

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Step 1.1

Multiply $\frac{\frac{x + 1}{x + 4} - \frac{1}{x}}{x + 1}$ by $\frac{(x + 4) x}{(x + 4) x}$ .

$\frac{(x + 4) x}{(x + 4) x} \cdot \frac{\frac{x + 1}{x + 4} - \frac{1}{x}}{x + 1}$

Step 1.2

Combine.

$\frac{(x + 4) x (\frac{x + 1}{x + 4} - \frac{1}{x})}{(x + 4) x (x + 1)}$

Step 2

Apply the distributive property.

$\frac{(x + 4) x \frac{x + 1}{x + 4} + (x + 4) x (- \frac{1}{x})}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 4$ .

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Step 3.1.1

Factor $x + 4$ out of $(x + 4) x$ .

$\frac{(x + 4) (x) \frac{x + 1}{x + 4} + (x + 4) x (- \frac{1}{x})}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 3.1.2

Cancel the common factor.

$\frac{(x + 4) x \frac{x + 1}{x + 4} + (x + 4) x (- \frac{1}{x})}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 3.1.3

Rewrite the expression.

$\frac{x (x + 1) + (x + 4) x (- \frac{1}{x})}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{x (x + 1) + (x + 4) x \frac{- 1}{x}}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 3.2.2

Factor $x$ out of $(x + 4) x$ .

$\frac{x (x + 1) + x (x + 4) \frac{- 1}{x}}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 3.2.3

Cancel the common factor.

$\frac{x (x + 1) + x (x + 4) \frac{- 1}{x}}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 3.2.4

Rewrite the expression.

$\frac{x (x + 1) + (x + 4) \cdot - 1}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 1 + (x + 4) \cdot - 1}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 4.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 1 + (x + 4) \cdot - 1}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 4.3

Multiply $x$ by $1$ .

$\frac{x^{2} + x + (x + 4) \cdot - 1}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 4.4

Apply the distributive property.

$\frac{x^{2} + x + x \cdot - 1 + 4 \cdot - 1}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 4.5

Move $- 1$ to the left of $x$ .

$\frac{x^{2} + x - 1 \cdot x + 4 \cdot - 1}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 4.6

Multiply $4$ by $- 1$ .

$\frac{x^{2} + x - 1 \cdot x - 4}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 4.7

Rewrite $- 1 x$ as $- x$ .

$\frac{x^{2} + x - x - 4}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 4.8

Subtract $x$ from $x$ .

$\frac{x^{2} + 0 - 4}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 4.9

Add $x^{2}$ and $0$ .

$\frac{x^{2} - 4}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 4.10

Rewrite $x^{2} - 4$ in a factored form.

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Step 4.10.1

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

$\frac{x^{2} - 2^{2}}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 4.10.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{(x + 2) (x - 2)}{(x + 4) x \cdot x + (x + 4) x \cdot 1}$

Step 5

Simplify the denominator.

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Step 5.1

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

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Step 5.1.1

Move $x$ .

$\frac{(x + 2) (x - 2)}{(x + 4) (x \cdot x) + (x + 4) x \cdot 1}$

Step 5.1.2

Multiply $x$ by $x$ .

$\frac{(x + 2) (x - 2)}{(x + 4) x^{2} + (x + 4) x \cdot 1}$

Step 5.2

Apply the distributive property.

$\frac{(x + 2) (x - 2)}{x \cdot x^{2} + 4 x^{2} + (x + 4) x \cdot 1}$

Step 5.3

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

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Step 5.3.1

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

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Step 5.3.1.1

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

$\frac{(x + 2) (x - 2)}{x^{1} x^{2} + 4 x^{2} + (x + 4) x \cdot 1}$

Step 5.3.1.2

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

$\frac{(x + 2) (x - 2)}{x^{1 + 2} + 4 x^{2} + (x + 4) x \cdot 1}$

Step 5.3.2

Add $1$ and $2$ .

$\frac{(x + 2) (x - 2)}{x^{3} + 4 x^{2} + (x + 4) x \cdot 1}$

Step 5.4

Apply the distributive property.

$\frac{(x + 2) (x - 2)}{x^{3} + 4 x^{2} + (x \cdot x + 4 x) \cdot 1}$

Step 5.5

Multiply $x$ by $x$ .

$\frac{(x + 2) (x - 2)}{x^{3} + 4 x^{2} + (x^{2} + 4 x) \cdot 1}$

Step 5.6

Multiply $x^{2} + 4 x$ by $1$ .

$\frac{(x + 2) (x - 2)}{x^{3} + 4 x^{2} + x^{2} + 4 x}$

Step 5.7

Add $4 x^{2}$ and $x^{2}$ .

$\frac{(x + 2) (x - 2)}{x^{3} + 5 x^{2} + 4 x}$

Step 5.8

Rewrite $x^{3} + 5 x^{2} + 4 x$ in a factored form.

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Step 5.8.1

Factor $x$ out of $x^{3} + 5 x^{2} + 4 x$ .

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Step 5.8.1.1

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

$\frac{(x + 2) (x - 2)}{x \cdot x^{2} + 5 x^{2} + 4 x}$

Step 5.8.1.2

Factor $x$ out of $5 x^{2}$ .

$\frac{(x + 2) (x - 2)}{x \cdot x^{2} + x (5 x) + 4 x}$

Step 5.8.1.3

Factor $x$ out of $4 x$ .

$\frac{(x + 2) (x - 2)}{x \cdot x^{2} + x (5 x) + x \cdot 4}$

Step 5.8.1.4

Factor $x$ out of $x \cdot x^{2} + x (5 x)$ .

$\frac{(x + 2) (x - 2)}{x (x^{2} + 5 x) + x \cdot 4}$

Step 5.8.1.5

Factor $x$ out of $x (x^{2} + 5 x) + x \cdot 4$ .

$\frac{(x + 2) (x - 2)}{x (x^{2} + 5 x + 4)}$

Step 5.8.2

Factor $x^{2} + 5 x + 4$ using the AC method.

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Step 5.8.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $4$ and whose sum is $5$ .

$1, 4$

Step 5.8.2.2

Write the factored form using these integers.

$\frac{(x + 2) (x - 2)}{x ((x + 1) (x + 4))}$

$\frac{(x + 2) (x - 2)}{x (x + 1) (x + 4)}$