Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

$\frac{4 x (x) + 4 x (- 1)}{2 x^{2} + 4 x - 6} \cdot \frac{x^{2} + x - 6}{4 x^{2} + 8 x}$

Step 1.3

Factor $4 x$ out of $4 x (x) + 4 x (- 1)$ .

$\frac{4 x (x - 1)}{2 x^{2} + 4 x - 6} \cdot \frac{x^{2} + x - 6}{4 x^{2} + 8 x}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2$ out of $2 x^{2} + 4 x - 6$ .

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Step 2.1.1

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

$\frac{4 x (x - 1)}{2 (x^{2}) + 4 x - 6} \cdot \frac{x^{2} + x - 6}{4 x^{2} + 8 x}$

Step 2.1.2

Factor $2$ out of $4 x$ .

$\frac{4 x (x - 1)}{2 (x^{2}) + 2 (2 x) - 6} \cdot \frac{x^{2} + x - 6}{4 x^{2} + 8 x}$

Step 2.1.3

Factor $2$ out of $- 6$ .

$\frac{4 x (x - 1)}{2 x^{2} + 2 (2 x) + 2 \cdot - 3} \cdot \frac{x^{2} + x - 6}{4 x^{2} + 8 x}$

Step 2.1.4

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

$\frac{4 x (x - 1)}{2 (x^{2} + 2 x) + 2 \cdot - 3} \cdot \frac{x^{2} + x - 6}{4 x^{2} + 8 x}$

Step 2.1.5

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

$\frac{4 x (x - 1)}{2 (x^{2} + 2 x - 3)} \cdot \frac{x^{2} + x - 6}{4 x^{2} + 8 x}$

Step 2.2

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

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Step 2.2.1

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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 2.2.2

Write the factored form using these integers.

$\frac{4 x (x - 1)}{2 ((x - 1) (x + 3))} \cdot \frac{x^{2} + x - 6}{4 x^{2} + 8 x}$

$\frac{4 x (x - 1)}{2 (x - 1) (x + 3)} \cdot \frac{x^{2} + x - 6}{4 x^{2} + 8 x}$

Step 3

Factor $x^{2} + x - 6$ using the AC method.

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Step 3.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 3.2

Write the factored form using these integers.

$\frac{4 x (x - 1)}{2 (x - 1) (x + 3)} \cdot \frac{(x - 2) (x + 3)}{4 x^{2} + 8 x}$

Step 4

Simplify terms.

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Step 4.1

Factor $4 x$ out of $4 x^{2} + 8 x$ .

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Step 4.1.1

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

$\frac{4 x (x - 1)}{2 (x - 1) (x + 3)} \cdot \frac{(x - 2) (x + 3)}{4 x (x) + 8 x}$

Step 4.1.2

Factor $4 x$ out of $8 x$ .

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

Step 4.1.3

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

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

Step 4.2

Cancel the common factor of $4 x$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $x + 3$ .

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Step 4.3.1

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

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

Step 4.3.2

Factor $x + 3$ out of $(x - 2) (x + 3)$ .

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

Step 4.3.3

Cancel the common factor.

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

Step 4.3.4

Rewrite the expression.

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

Step 4.4

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

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

Step 4.5

Cancel the common factor of $x - 1$ .

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Rewrite the expression.

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