Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $3 x$ .

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Step 1.1

Multiply $\frac{x - \frac{x^{2}}{3}}{\frac{x}{3} - \frac{3}{x}}$ by $\frac{3 x}{3 x}$ .

$\frac{3 x}{3 x} \cdot \frac{x - \frac{x^{2}}{3}}{\frac{x}{3} - \frac{3}{x}}$

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $3$ .

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Step 3.1.1

Move the leading negative in $- \frac{x^{2}}{3}$ into the numerator.

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

Step 3.1.2

Factor $3$ out of $3 x$ .

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

Step 3.1.3

Cancel the common factor.

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

Step 3.1.4

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

Cancel the common factor of $3$ .

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Step 3.5.1

Factor $3$ out of $3 x$ .

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

Step 3.5.2

Cancel the common factor.

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

Step 3.5.3

Rewrite the expression.

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Add $1$ and $1$ .

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

Step 3.10

Cancel the common factor of $x$ .

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Step 3.10.1

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

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

Step 3.10.2

Factor $x$ out of $3 x$ .

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

Step 3.10.3

Cancel the common factor.

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

Step 3.10.4

Rewrite the expression.

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

Step 3.11

Multiply $3$ by $- 3$ .

$\frac{3 x \cdot x - x^{3}}{x^{2} - 9}$

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Factor $x^{2}$ out of $3 x \cdot x$ .

$\frac{x^{2} (3 x^{- 1} x) - x^{3}}{x^{2} - 9}$

Step 4.1.2

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

$\frac{x^{2} (3 x^{- 1} x) + x^{2} (- x)}{x^{2} - 9}$

Step 4.1.3

Factor $x^{2}$ out of $x^{2} (3 x^{- 1} x) + x^{2} (- x)$ .

$\frac{x^{2} (3 x^{- 1} x - x)}{x^{2} - 9}$

Step 4.2

Multiply $x^{- 1}$ by $x$ by adding the exponents.

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Step 4.2.1

Move $x$ .

$\frac{x^{2} (3 (x \cdot x^{- 1}) - x)}{x^{2} - 9}$

Step 4.2.2

Multiply $x$ by $x^{- 1}$ .

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Step 4.2.2.1

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

$\frac{x^{2} (3 (x^{1} x^{- 1}) - x)}{x^{2} - 9}$

Step 4.2.2.2

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

$\frac{x^{2} (3 x^{1 - 1} - x)}{x^{2} - 9}$

Step 4.2.3

Subtract $1$ from $1$ .

$\frac{x^{2} (3 x^{0} - x)}{x^{2} - 9}$

Step 4.3

Simplify $3 x^{0}$ .

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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 $b = 3$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

Reorder terms.

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

Step 6.1.5

Cancel the common factor.

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

Step 6.1.6

Rewrite the expression.

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

Step 6.2

Simplify the expression.

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Step 6.2.1

Move $- 1$ to the left of $x^{2}$ .

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

Step 6.2.2

Move the negative in front of the fraction.

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