Precalculus Examples

$\frac{3}{x} - \frac{6}{x + 3} - \frac{x}{{(x + 3)}^{2}}$

Step 1

Find the common denominator.

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Step 1.1

Multiply $\frac{3}{x}$ by $\frac{{(x + 3)}^{2}}{{(x + 3)}^{2}}$ .

$\frac{3}{x} \cdot \frac{{(x + 3)}^{2}}{{(x + 3)}^{2}} - \frac{6}{x + 3} - \frac{x}{{(x + 3)}^{2}}$

Step 1.2

Multiply $\frac{3}{x}$ by $\frac{{(x + 3)}^{2}}{{(x + 3)}^{2}}$ .

$\frac{3 {(x + 3)}^{2}}{x {(x + 3)}^{2}} - \frac{6}{x + 3} - \frac{x}{{(x + 3)}^{2}}$

Step 1.3

Multiply $\frac{6}{x + 3}$ by $\frac{x (x + 3)}{x (x + 3)}$ .

$\frac{3 {(x + 3)}^{2}}{x {(x + 3)}^{2}} - (\frac{6}{x + 3} \cdot \frac{x (x + 3)}{x (x + 3)}) - \frac{x}{{(x + 3)}^{2}}$

Step 1.4

Multiply $\frac{6}{x + 3}$ by $\frac{x (x + 3)}{x (x + 3)}$ .

$\frac{3 {(x + 3)}^{2}}{x {(x + 3)}^{2}} - \frac{6 (x (x + 3))}{(x + 3) (x (x + 3))} - \frac{x}{{(x + 3)}^{2}}$

Step 1.5

Multiply $\frac{x}{{(x + 3)}^{2}}$ by $\frac{x}{x}$ .

$\frac{3 {(x + 3)}^{2}}{x {(x + 3)}^{2}} - \frac{6 (x (x + 3))}{(x + 3) (x (x + 3))} - (\frac{x}{{(x + 3)}^{2}} \cdot \frac{x}{x})$

Step 1.6

Multiply $\frac{x}{{(x + 3)}^{2}}$ by $\frac{x}{x}$ .

$\frac{3 {(x + 3)}^{2}}{x {(x + 3)}^{2}} - \frac{6 (x (x + 3))}{(x + 3) (x (x + 3))} - \frac{x \cdot x}{{(x + 3)}^{2} x}$

Step 1.7

Reorder the factors of $(x + 3) (x (x + 3))$ .

$\frac{3 {(x + 3)}^{2}}{x {(x + 3)}^{2}} - \frac{6 (x (x + 3))}{x (x + 3) (x + 3)} - \frac{x \cdot x}{{(x + 3)}^{2} x}$

Step 1.8

Raise $x + 3$ to the power of $1$ .

$\frac{3 {(x + 3)}^{2}}{x {(x + 3)}^{2}} - \frac{6 (x (x + 3))}{x ({(x + 3)}^{1} (x + 3))} - \frac{x \cdot x}{{(x + 3)}^{2} x}$

Step 1.9

Raise $x + 3$ to the power of $1$ .

$\frac{3 {(x + 3)}^{2}}{x {(x + 3)}^{2}} - \frac{6 (x (x + 3))}{x ({(x + 3)}^{1} {(x + 3)}^{1})} - \frac{x \cdot x}{{(x + 3)}^{2} x}$

Step 1.10

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

$\frac{3 {(x + 3)}^{2}}{x {(x + 3)}^{2}} - \frac{6 (x (x + 3))}{x {(x + 3)}^{1 + 1}} - \frac{x \cdot x}{{(x + 3)}^{2} x}$

Step 1.11

Add $1$ and $1$ .

$\frac{3 {(x + 3)}^{2}}{x {(x + 3)}^{2}} - \frac{6 (x (x + 3))}{x {(x + 3)}^{2}} - \frac{x \cdot x}{{(x + 3)}^{2} x}$

Step 1.12

Reorder the factors of ${(x + 3)}^{2} x$ .

$\frac{3 {(x + 3)}^{2}}{x {(x + 3)}^{2}} - \frac{6 (x (x + 3))}{x {(x + 3)}^{2}} - \frac{x \cdot x}{x {(x + 3)}^{2}}$

Step 2

Combine the numerators over the common denominator.

$\frac{3 {(x + 3)}^{2} - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$\frac{3 ((x + 3) (x + 3)) - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$\frac{3 (x (x + 3) + 3 (x + 3)) - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.2.2

Apply the distributive property.

$\frac{3 (x \cdot x + x \cdot 3 + 3 (x + 3)) - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.2.3

Apply the distributive property.

$\frac{3 (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

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 $x$ .

$\frac{3 (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.3.1.2

Move $3$ to the left of $x$ .

$\frac{3 (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.3.1.3

Multiply $3$ by $3$ .

$\frac{3 (x^{2} + 3 x + 3 x + 9) - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.3.2

Add $3 x$ and $3 x$ .

$\frac{3 (x^{2} + 6 x + 9) - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.4

Apply the distributive property.

$\frac{3 x^{2} + 3 (6 x) + 3 \cdot 9 - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.5

Simplify.

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Step 3.5.1

Multiply $6$ by $3$ .

$\frac{3 x^{2} + 18 x + 3 \cdot 9 - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.5.2

Multiply $3$ by $9$ .

$\frac{3 x^{2} + 18 x + 27 - 6 (x (x + 3)) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.6

Apply the distributive property.

$\frac{3 x^{2} + 18 x + 27 - 6 (x \cdot x + x \cdot 3) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.7

Multiply $x$ by $x$ .

$\frac{3 x^{2} + 18 x + 27 - 6 (x^{2} + x \cdot 3) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.8

Move $3$ to the left of $x$ .

$\frac{3 x^{2} + 18 x + 27 - 6 (x^{2} + 3 \cdot x) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.9

Apply the distributive property.

$\frac{3 x^{2} + 18 x + 27 - 6 x^{2} - 6 (3 x) - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.10

Multiply $3$ by $- 6$ .

$\frac{3 x^{2} + 18 x + 27 - 6 x^{2} - 18 x - x \cdot x}{x {(x + 3)}^{2}}$

Step 3.11

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

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Step 3.11.1

Move $x$ .

$\frac{3 x^{2} + 18 x + 27 - 6 x^{2} - 18 x - (x \cdot x)}{x {(x + 3)}^{2}}$

Step 3.11.2

Multiply $x$ by $x$ .

$\frac{3 x^{2} + 18 x + 27 - 6 x^{2} - 18 x - x^{2}}{x {(x + 3)}^{2}}$

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $3 x^{2} + 18 x + 27 - 6 x^{2} - 18 x - x^{2}$ .

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Step 4.1.1

Subtract $18 x$ from $18 x$ .

$\frac{3 x^{2} + 27 - 6 x^{2} + 0 - x^{2}}{x {(x + 3)}^{2}}$

Step 4.1.2

Add $3 x^{2} + 27 - 6 x^{2}$ and $0$ .

$\frac{3 x^{2} + 27 - 6 x^{2} - x^{2}}{x {(x + 3)}^{2}}$

Step 4.2

Subtract $6 x^{2}$ from $3 x^{2}$ .

$\frac{- 3 x^{2} + 27 - x^{2}}{x {(x + 3)}^{2}}$

Step 4.3

Subtract $x^{2}$ from $- 3 x^{2}$ .

$\frac{- 4 x^{2} + 27}{x {(x + 3)}^{2}}$

Step 4.4

Factor $- 1$ out of $- 4 x^{2}$ .

$\frac{- (4 x^{2}) + 27}{x {(x + 3)}^{2}}$

Step 4.5

Rewrite $27$ as $- 1 (- 27)$ .

$\frac{- (4 x^{2}) - 1 (- 27)}{x {(x + 3)}^{2}}$

Step 4.6

Factor $- 1$ out of $- (4 x^{2}) - 1 (- 27)$ .

$\frac{- (4 x^{2} - 27)}{x {(x + 3)}^{2}}$

Step 4.7

Simplify the expression.

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Step 4.7.1

Rewrite $- (4 x^{2} - 27)$ as $- 1 (4 x^{2} - 27)$ .

$\frac{- 1 (4 x^{2} - 27)}{x {(x + 3)}^{2}}$

Step 4.7.2

Move the negative in front of the fraction.

$- \frac{4 x^{2} - 27}{x {(x + 3)}^{2}}$