Algebra Examples

$\frac{\frac{2}{x + 3} - \frac{1}{3}}{\frac{x^{2}}{3} - 3}$

Step 1

Multiply the numerator and denominator of the fraction by $(x + 3) \cdot 3$ .

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Step 1.1

Multiply $\frac{\frac{2}{x + 3} - \frac{1}{3}}{\frac{x^{2}}{3} - 3}$ by $\frac{(x + 3) \cdot 3}{(x + 3) \cdot 3}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 3$ .

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Step 3.1.1

Factor $x + 3$ out of $(x + 3) \cdot 3$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $3$ by $2$ .

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

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Step 3.3.2

Factor $3$ out of $(x + 3) \cdot 3$ .

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $3$ .

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Step 3.4.1

Factor $3$ out of $(x + 3) \cdot 3$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

Multiply $3$ by $- 1$ .

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

Step 4.4

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

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

Step 4.5

Subtract $3$ from $6$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $x + 3$ out of $(x + 3) x^{2} + (x + 3) \cdot 3 \cdot - 3$ .

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Step 5.1.1

Factor $x + 3$ out of $(x + 3) x^{2}$ .

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

Step 5.1.2

Factor $x + 3$ out of $(x + 3) \cdot 3 \cdot - 3$ .

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

Step 5.1.3

Factor $x + 3$ out of $(x + 3) (x^{2}) + (x + 3) (3 \cdot - 3)$ .

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

Step 5.2

Multiply $3$ by $- 3$ .

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

Step 5.3

Rewrite $x^{2} - 9$ in a factored form.

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Step 5.3.1

Rewrite $9$ as $3^{2}$ .

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

Step 5.3.2

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 $b = 3$ .

$\frac{- x + 3}{(x + 3) ((x + 3) (x - 3))}$

$\frac{- x + 3}{(x + 3) (x + 3) (x - 3)}$

Step 5.4

Combine exponents.

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

Add $1$ and $1$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $- x + 3$ and $x - 3$ .

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Step 6.1.1

Factor $- 1$ out of $- x$ .

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

Step 6.1.2

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 6.1.3

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

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

Step 6.1.4

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

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

Step 6.1.5

Cancel the common factor.

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

Step 6.1.6

Rewrite the expression.

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

Step 6.2

Move the negative in front of the fraction.

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