Algebra Examples

$\frac{x^{2} - 4 x}{x - 1} \cdot \frac{x^{2} + 3 x - 4}{2 x}$

Step 1

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

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Step 1.1

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

$\frac{x \cdot x - 4 x}{x - 1} \cdot \frac{x^{2} + 3 x - 4}{2 x}$

Step 1.2

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

$\frac{x \cdot x + x \cdot - 4}{x - 1} \cdot \frac{x^{2} + 3 x - 4}{2 x}$

Step 1.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

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

Step 2

Factor $x^{2} + 3 x - 4$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 2.2

Write the factored form using these integers.

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Factor $x$ out of $2 x$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Simplify terms.

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Step 3.2.1

Cancel the common factor of $x - 1$ .

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Step 3.2.1.1

Factor $x^{2} + 3 x - 4$ using the AC method.

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Step 3.2.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 3.2.1.1.2

Write the factored form using these integers.

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

Step 3.2.1.2

Cancel the common factor.

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

Step 3.2.1.3

Rewrite the expression.

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Combine $x$ and $\frac{x + 4}{2}$ .

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

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Factor $2$ out of $- 4$ .

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

Step 3.2.4.2

Cancel the common factor.

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

Step 3.2.4.3

Rewrite the expression.

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

Step 3.3

Simplify each term.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Multiply $- 2$ by $4$ .

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

Step 4

Find the common denominator.

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Step 4.1

Write $- 2 x$ as a fraction with denominator $1$ .

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

Step 4.2

Multiply $\frac{- 2 x}{1}$ by $\frac{2}{2}$ .

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

Step 4.3

Multiply $\frac{- 2 x}{1}$ by $\frac{2}{2}$ .

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

Step 4.4

Write $- 8$ as a fraction with denominator $1$ .

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

Step 4.5

Multiply $\frac{- 8}{1}$ by $\frac{2}{2}$ .

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

Step 4.6

Multiply $\frac{- 8}{1}$ by $\frac{2}{2}$ .

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

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Multiply $x$ by $x$ .

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

Step 5.2.3

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

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

Step 5.2.4

Multiply $2$ by $- 2$ .

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

Step 5.2.5

Multiply $- 8$ by $2$ .

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

Step 5.3

Combine the opposite terms in $x^{2} + 4 x - 4 x - 16$ .

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Step 5.3.1

Subtract $4 x$ from $4 x$ .

$\frac{x^{2} + 0 - 16}{2}$

Step 5.3.2

Add $x^{2}$ and $0$ .

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

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $16$ as $4^{2}$ .

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

Step 6.2

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 $b = 4$ .

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