Algebra Examples

$\frac{\frac{4}{x - 2} - \frac{x}{16}}{\frac{x^{2}}{2 x - 4} + \frac{x - 2}{4}}$

Step 1

Multiply the numerator and denominator of the fraction by $(x - 2) \cdot 16 (2 x - 4)$ .

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Step 1.1

Multiply $\frac{\frac{4}{x - 2} - \frac{x}{16}}{\frac{x^{2}}{2 x - 4} + \frac{x - 2}{4}}$ by $\frac{(x - 2) \cdot 16 (2 x - 4)}{(x - 2) \cdot 16 (2 x - 4)}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x - 2$ .

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Step 3.1.1

Factor $x - 2$ out of $(x - 2) \cdot 16 (2 x - 4)$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $4$ by $16$ .

$\frac{64 (2 x - 4) + (x - 2) \cdot 16 (2 x - 4) (- \frac{x}{16})}{(x - 2) \cdot 16 (2 x - 4) \frac{x^{2}}{2 x - 4} + (x - 2) \cdot 16 (2 x - 4) \frac{x - 2}{4}}$

Step 3.3

Cancel the common factor of $16$ .

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Step 3.3.1

Move the leading negative in $- \frac{x}{16}$ into the numerator.

$\frac{64 (2 x - 4) + (x - 2) \cdot 16 (2 x - 4) \frac{- x}{16}}{(x - 2) \cdot 16 (2 x - 4) \frac{x^{2}}{2 x - 4} + (x - 2) \cdot 16 (2 x - 4) \frac{x - 2}{4}}$

Step 3.3.2

Factor $16$ out of $(x - 2) \cdot 16 (2 x - 4)$ .

$\frac{64 (2 x - 4) + 16 ((x - 2) (2 x - 4)) \frac{- x}{16}}{(x - 2) \cdot 16 (2 x - 4) \frac{x^{2}}{2 x - 4} + (x - 2) \cdot 16 (2 x - 4) \frac{x - 2}{4}}$

Step 3.3.3

Cancel the common factor.

$\frac{64 (2 x - 4) + 16 ((x - 2) (2 x - 4)) \frac{- x}{16}}{(x - 2) \cdot 16 (2 x - 4) \frac{x^{2}}{2 x - 4} + (x - 2) \cdot 16 (2 x - 4) \frac{x - 2}{4}}$

Step 3.3.4

Rewrite the expression.

$\frac{64 (2 x - 4) + (x - 2) (2 x - 4) (- x)}{(x - 2) \cdot 16 (2 x - 4) \frac{x^{2}}{2 x - 4} + (x - 2) \cdot 16 (2 x - 4) \frac{x - 2}{4}}$

Step 3.4

Cancel the common factor of $2 x - 4$ .

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Step 3.4.1

Factor $2 x - 4$ out of $(x - 2) \cdot 16 (2 x - 4)$ .

$\frac{64 (2 x - 4) + (x - 2) (2 x - 4) (- x)}{(2 x - 4) ((x - 2) \cdot 16) \frac{x^{2}}{2 x - 4} + (x - 2) \cdot 16 (2 x - 4) \frac{x - 2}{4}}$

Step 3.4.2

Cancel the common factor.

$\frac{64 (2 x - 4) + (x - 2) (2 x - 4) (- x)}{(2 x - 4) ((x - 2) \cdot 16) \frac{x^{2}}{2 x - 4} + (x - 2) \cdot 16 (2 x - 4) \frac{x - 2}{4}}$

Step 3.4.3

Rewrite the expression.

$\frac{64 (2 x - 4) + (x - 2) (2 x - 4) (- x)}{(x - 2) \cdot 16 x^{2} + (x - 2) \cdot 16 (2 x - 4) \frac{x - 2}{4}}$

Step 3.5

Cancel the common factor of $4$ .

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Step 3.5.1

Factor $4$ out of $(x - 2) \cdot 16 (2 x - 4)$ .

$\frac{64 (2 x - 4) + (x - 2) (2 x - 4) (- x)}{(x - 2) \cdot 16 x^{2} + 4 ((x - 2) \cdot 4 (2 x - 4)) \frac{x - 2}{4}}$

Step 3.5.2

Cancel the common factor.

$\frac{64 (2 x - 4) + (x - 2) (2 x - 4) (- x)}{(x - 2) \cdot 16 x^{2} + 4 ((x - 2) \cdot 4 (2 x - 4)) \frac{x - 2}{4}}$

Step 3.5.3

Rewrite the expression.

$\frac{64 (2 x - 4) + (x - 2) (2 x - 4) (- x)}{(x - 2) \cdot 16 x^{2} + (x - 2) \cdot 4 (2 x - 4) (x - 2)}$

Step 3.6

Raise $x - 2$ to the power of $1$ .

$\frac{64 (2 x - 4) + (x - 2) (2 x - 4) (- x)}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) ({(x - 2)}^{1} (x - 2))}$

Step 3.7

Raise $x - 2$ to the power of $1$ .

$\frac{64 (2 x - 4) + (x - 2) (2 x - 4) (- x)}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) ({(x - 2)}^{1} {(x - 2)}^{1})}$

Step 3.8

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

$\frac{64 (2 x - 4) + (x - 2) (2 x - 4) (- x)}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{1 + 1}}$

Step 3.9

Add $1$ and $1$ .

$\frac{64 (2 x - 4) + (x - 2) (2 x - 4) (- x)}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $2 x - 4$ out of $64 (2 x - 4) + (x - 2) (2 x - 4) (- x)$ .

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Step 4.1.1

Factor $2 x - 4$ out of $64 (2 x - 4)$ .

$\frac{(2 x - 4) \cdot 64 + (x - 2) (2 x - 4) (- x)}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4.1.2

Factor $2 x - 4$ out of $(x - 2) (2 x - 4) (- x)$ .

$\frac{(2 x - 4) \cdot 64 + (2 x - 4) ((x - 2) (- x))}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4.1.3

Factor $2 x - 4$ out of $(2 x - 4) \cdot 64 + (2 x - 4) ((x - 2) (- x))$ .

$\frac{(2 x - 4) (64 + (x - 2) (- x))}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4.2

Factor $2$ out of $2 x - 4$ .

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Step 4.2.1

Factor $2$ out of $2 x$ .

$\frac{(2 (x) - 4) (64 + (x - 2) (- x))}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4.2.2

Factor $2$ out of $- 4$ .

$\frac{(2 x + 2 \cdot - 2) (64 + (x - 2) (- x))}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4.2.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$\frac{2 (x - 2) (64 + (x - 2) (- x))}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4.3

Apply the distributive property.

$\frac{2 (x - 2) (64 + x (- x) - 2 (- x))}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4.4

Rewrite using the commutative property of multiplication.

$\frac{2 (x - 2) (64 - x \cdot x - 2 (- x))}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4.5

Multiply $- 1$ by $- 2$ .

$\frac{2 (x - 2) (64 - x \cdot x + 2 x)}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4.6

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

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Step 4.6.1

Move $x$ .

$\frac{2 (x - 2) (64 - (x \cdot x) + 2 x)}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4.6.2

Multiply $x$ by $x$ .

$\frac{2 (x - 2) (64 - x^{2} + 2 x)}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 4.7

Reorder terms.

$\frac{2 (x - 2) (- x^{2} + 2 x + 64)}{(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $(x - 2) \cdot 4$ out of $(x - 2) \cdot 16 x^{2} + 4 (2 x - 4) {(x - 2)}^{2}$ .

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Step 5.1.1

Factor $(x - 2) \cdot 4$ out of $(x - 2) \cdot 16 x^{2}$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Expand $(2 x - 4) (x - 2)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

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

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Step 5.3.1.1.1

Move $x$ .

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

Step 5.3.1.1.2

Multiply $x$ by $x$ .

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

Step 5.3.1.2

Multiply $- 2$ by $2$ .

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

Step 5.3.1.3

Multiply $- 4$ by $- 2$ .

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

Step 5.3.2

Subtract $4 x$ from $- 4 x$ .

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

Step 5.4

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

$\frac{2 (x - 2) (- x^{2} + 2 x + 64)}{(x - 2) \cdot 4 (6 x^{2} - 8 x + 8)}$

Step 5.5

Factor $2$ out of $6 x^{2} - 8 x + 8$ .

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Step 5.5.1

Factor $2$ out of $6 x^{2}$ .

$\frac{2 (x - 2) (- x^{2} + 2 x + 64)}{(x - 2) \cdot 4 (2 (3 x^{2}) - 8 x + 8)}$

Step 5.5.2

Factor $2$ out of $- 8 x$ .

$\frac{2 (x - 2) (- x^{2} + 2 x + 64)}{(x - 2) \cdot 4 (2 (3 x^{2}) + 2 (- 4 x) + 8)}$

Step 5.5.3

Factor $2$ out of $8$ .

$\frac{2 (x - 2) (- x^{2} + 2 x + 64)}{(x - 2) \cdot 4 (2 (3 x^{2}) + 2 (- 4 x) + 2 (4))}$

Step 5.5.4

Factor $2$ out of $2 (3 x^{2}) + 2 (- 4 x)$ .

$\frac{2 (x - 2) (- x^{2} + 2 x + 64)}{(x - 2) \cdot 4 (2 (3 x^{2} - 4 x) + 2 (4))}$

Step 5.5.5

Factor $2$ out of $2 (3 x^{2} - 4 x) + 2 (4)$ .

$\frac{2 (x - 2) (- x^{2} + 2 x + 64)}{(x - 2) \cdot 4 (2 (3 x^{2} - 4 x + 4))}$

$\frac{2 (x - 2) (- x^{2} + 2 x + 64)}{(x - 2) \cdot 4 \cdot 2 (3 x^{2} - 4 x + 4)}$

Step 5.6

Multiply $2$ by $4$ .

$\frac{2 (x - 2) (- x^{2} + 2 x + 64)}{(x - 2) \cdot 8 (3 x^{2} - 4 x + 4)}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $2$ and $8$ .

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Step 6.1.1

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

$\frac{2 ((x - 2) (- x^{2} + 2 x + 64))}{(x - 2) \cdot 8 (3 x^{2} - 4 x + 4)}$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $2$ out of $(x - 2) \cdot 8 (3 x^{2} - 4 x + 4)$ .

$\frac{2 ((x - 2) (- x^{2} + 2 x + 64))}{2 ((x - 2) \cdot 4 (3 x^{2} - 4 x + 4))}$

Step 6.1.2.2

Cancel the common factor.

$\frac{2 ((x - 2) (- x^{2} + 2 x + 64))}{2 ((x - 2) \cdot 4 (3 x^{2} - 4 x + 4))}$

Step 6.1.2.3

Rewrite the expression.

$\frac{(x - 2) (- x^{2} + 2 x + 64)}{(x - 2) \cdot 4 (3 x^{2} - 4 x + 4)}$

Step 6.2

Cancel the common factor of $x - 2$ .

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Step 6.2.1

Cancel the common factor.

$\frac{(x - 2) (- x^{2} + 2 x + 64)}{(x - 2) \cdot 4 (3 x^{2} - 4 x + 4)}$

Step 6.2.2

Rewrite the expression.

$\frac{- x^{2} + 2 x + 64}{4 (3 x^{2} - 4 x + 4)}$

Step 6.3

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

$\frac{- (x^{2}) + 2 x + 64}{4 (3 x^{2} - 4 x + 4)}$

Step 6.4

Factor $- 1$ out of $2 x$ .

$\frac{- (x^{2}) - (- 2 x) + 64}{4 (3 x^{2} - 4 x + 4)}$

Step 6.5

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

$\frac{- (x^{2} - 2 x) + 64}{4 (3 x^{2} - 4 x + 4)}$

Step 6.6

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

$\frac{- (x^{2} - 2 x) - 1 (- 64)}{4 (3 x^{2} - 4 x + 4)}$

Step 6.7

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

$\frac{- (x^{2} - 2 x - 64)}{4 (3 x^{2} - 4 x + 4)}$

Step 6.8

Simplify the expression.

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Step 6.8.1

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

$\frac{- 1 (x^{2} - 2 x - 64)}{4 (3 x^{2} - 4 x + 4)}$

Step 6.8.2

Move the negative in front of the fraction.

$- \frac{x^{2} - 2 x - 64}{4 (3 x^{2} - 4 x + 4)}$