Precalculus Examples

$\frac{4}{x} - \frac{8}{x + 4} - \frac{x}{{(x + 4)}^{2}}$

Step 1

Find the common denominator.

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Step 1.1

Multiply $\frac{4}{x}$ by $\frac{{(x + 4)}^{2}}{{(x + 4)}^{2}}$ .

$\frac{4}{x} \cdot \frac{{(x + 4)}^{2}}{{(x + 4)}^{2}} - \frac{8}{x + 4} - \frac{x}{{(x + 4)}^{2}}$

Step 1.2

Multiply $\frac{4}{x}$ by $\frac{{(x + 4)}^{2}}{{(x + 4)}^{2}}$ .

$\frac{4 {(x + 4)}^{2}}{x {(x + 4)}^{2}} - \frac{8}{x + 4} - \frac{x}{{(x + 4)}^{2}}$

Step 1.3

Multiply $\frac{8}{x + 4}$ by $\frac{x (x + 4)}{x (x + 4)}$ .

$\frac{4 {(x + 4)}^{2}}{x {(x + 4)}^{2}} - (\frac{8}{x + 4} \cdot \frac{x (x + 4)}{x (x + 4)}) - \frac{x}{{(x + 4)}^{2}}$

Step 1.4

Multiply $\frac{8}{x + 4}$ by $\frac{x (x + 4)}{x (x + 4)}$ .

$\frac{4 {(x + 4)}^{2}}{x {(x + 4)}^{2}} - \frac{8 (x (x + 4))}{(x + 4) (x (x + 4))} - \frac{x}{{(x + 4)}^{2}}$

Step 1.5

Multiply $\frac{x}{{(x + 4)}^{2}}$ by $\frac{x}{x}$ .

$\frac{4 {(x + 4)}^{2}}{x {(x + 4)}^{2}} - \frac{8 (x (x + 4))}{(x + 4) (x (x + 4))} - (\frac{x}{{(x + 4)}^{2}} \cdot \frac{x}{x})$

Step 1.6

Multiply $\frac{x}{{(x + 4)}^{2}}$ by $\frac{x}{x}$ .

$\frac{4 {(x + 4)}^{2}}{x {(x + 4)}^{2}} - \frac{8 (x (x + 4))}{(x + 4) (x (x + 4))} - \frac{x \cdot x}{{(x + 4)}^{2} x}$

Step 1.7

Reorder the factors of $(x + 4) (x (x + 4))$ .

$\frac{4 {(x + 4)}^{2}}{x {(x + 4)}^{2}} - \frac{8 (x (x + 4))}{x (x + 4) (x + 4)} - \frac{x \cdot x}{{(x + 4)}^{2} x}$

Step 1.8

Raise $x + 4$ to the power of $1$ .

$\frac{4 {(x + 4)}^{2}}{x {(x + 4)}^{2}} - \frac{8 (x (x + 4))}{x ({(x + 4)}^{1} (x + 4))} - \frac{x \cdot x}{{(x + 4)}^{2} x}$

Step 1.9

Raise $x + 4$ to the power of $1$ .

$\frac{4 {(x + 4)}^{2}}{x {(x + 4)}^{2}} - \frac{8 (x (x + 4))}{x ({(x + 4)}^{1} {(x + 4)}^{1})} - \frac{x \cdot x}{{(x + 4)}^{2} x}$

Step 1.10

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

$\frac{4 {(x + 4)}^{2}}{x {(x + 4)}^{2}} - \frac{8 (x (x + 4))}{x {(x + 4)}^{1 + 1}} - \frac{x \cdot x}{{(x + 4)}^{2} x}$

Step 1.11

Add $1$ and $1$ .

$\frac{4 {(x + 4)}^{2}}{x {(x + 4)}^{2}} - \frac{8 (x (x + 4))}{x {(x + 4)}^{2}} - \frac{x \cdot x}{{(x + 4)}^{2} x}$

Step 1.12

Reorder the factors of ${(x + 4)}^{2} x$ .

$\frac{4 {(x + 4)}^{2}}{x {(x + 4)}^{2}} - \frac{8 (x (x + 4))}{x {(x + 4)}^{2}} - \frac{x \cdot x}{x {(x + 4)}^{2}}$

Step 2

Combine the numerators over the common denominator.

$\frac{4 {(x + 4)}^{2} - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$\frac{4 ((x + 4) (x + 4)) - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$\frac{4 (x (x + 4) + 4 (x + 4)) - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.2.2

Apply the distributive property.

$\frac{4 (x \cdot x + x \cdot 4 + 4 (x + 4)) - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.2.3

Apply the distributive property.

$\frac{4 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $x$ by $x$ .

$\frac{4 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.3.1.2

Move $4$ to the left of $x$ .

$\frac{4 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.3.1.3

Multiply $4$ by $4$ .

$\frac{4 (x^{2} + 4 x + 4 x + 16) - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.3.2

Add $4 x$ and $4 x$ .

$\frac{4 (x^{2} + 8 x + 16) - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.4

Apply the distributive property.

$\frac{4 x^{2} + 4 (8 x) + 4 \cdot 16 - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.5

Simplify.

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Step 3.5.1

Multiply $8$ by $4$ .

$\frac{4 x^{2} + 32 x + 4 \cdot 16 - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.5.2

Multiply $4$ by $16$ .

$\frac{4 x^{2} + 32 x + 64 - 8 (x (x + 4)) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.6

Apply the distributive property.

$\frac{4 x^{2} + 32 x + 64 - 8 (x \cdot x + x \cdot 4) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.7

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 32 x + 64 - 8 (x^{2} + x \cdot 4) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.8

Move $4$ to the left of $x$ .

$\frac{4 x^{2} + 32 x + 64 - 8 (x^{2} + 4 \cdot x) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.9

Apply the distributive property.

$\frac{4 x^{2} + 32 x + 64 - 8 x^{2} - 8 (4 x) - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.10

Multiply $4$ by $- 8$ .

$\frac{4 x^{2} + 32 x + 64 - 8 x^{2} - 32 x - x \cdot x}{x {(x + 4)}^{2}}$

Step 3.11

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

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Step 3.11.1

Move $x$ .

$\frac{4 x^{2} + 32 x + 64 - 8 x^{2} - 32 x - (x \cdot x)}{x {(x + 4)}^{2}}$

Step 3.11.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 32 x + 64 - 8 x^{2} - 32 x - x^{2}}{x {(x + 4)}^{2}}$

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $4 x^{2} + 32 x + 64 - 8 x^{2} - 32 x - x^{2}$ .

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Step 4.1.1

Subtract $32 x$ from $32 x$ .

$\frac{4 x^{2} + 64 - 8 x^{2} + 0 - x^{2}}{x {(x + 4)}^{2}}$

Step 4.1.2

Add $4 x^{2} + 64 - 8 x^{2}$ and $0$ .

$\frac{4 x^{2} + 64 - 8 x^{2} - x^{2}}{x {(x + 4)}^{2}}$

Step 4.2

Subtract $8 x^{2}$ from $4 x^{2}$ .

$\frac{- 4 x^{2} + 64 - x^{2}}{x {(x + 4)}^{2}}$

Step 4.3

Subtract $x^{2}$ from $- 4 x^{2}$ .

$\frac{- 5 x^{2} + 64}{x {(x + 4)}^{2}}$

Step 4.4

Factor $- 1$ out of $- 5 x^{2}$ .

$\frac{- (5 x^{2}) + 64}{x {(x + 4)}^{2}}$

Step 4.5

Rewrite $64$ as $- 1 (- 64)$ .

$\frac{- (5 x^{2}) - 1 (- 64)}{x {(x + 4)}^{2}}$

Step 4.6

Factor $- 1$ out of $- (5 x^{2}) - 1 (- 64)$ .

$\frac{- (5 x^{2} - 64)}{x {(x + 4)}^{2}}$

Step 4.7

Simplify the expression.

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Step 4.7.1

Rewrite $- (5 x^{2} - 64)$ as $- 1 (5 x^{2} - 64)$ .

$\frac{- 1 (5 x^{2} - 64)}{x {(x + 4)}^{2}}$

Step 4.7.2

Move the negative in front of the fraction.

$- \frac{5 x^{2} - 64}{x {(x + 4)}^{2}}$