Algebra Examples

$\frac{x^{2} - 2 x - 8}{x^{2} - 2 x - 15} \div \frac{2 x^{2} - 8 x}{2 x^{2} - 10 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 2 x - 8}{x^{2} - 2 x - 15} \cdot \frac{2 x^{2} - 10 x}{2 x^{2} - 8 x}$

Step 2

Factor $x^{2} - 2 x - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 2)}{x^{2} - 2 x - 15} \cdot \frac{2 x^{2} - 10 x}{2 x^{2} - 8 x}$

Step 3

Factor $x^{2} - 2 x - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 2)}{(x - 5) (x + 3)} \cdot \frac{2 x^{2} - 10 x}{2 x^{2} - 8 x}$

Step 4

Simplify terms.

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Step 4.1

Factor $2 x$ out of $2 x^{2} - 10 x$ .

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Step 4.1.1

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

$\frac{(x - 4) (x + 2)}{(x - 5) (x + 3)} \cdot \frac{2 x (x) - 10 x}{2 x^{2} - 8 x}$

Step 4.1.2

Factor $2 x$ out of $- 10 x$ .

$\frac{(x - 4) (x + 2)}{(x - 5) (x + 3)} \cdot \frac{2 x (x) + 2 x (- 5)}{2 x^{2} - 8 x}$

Step 4.1.3

Factor $2 x$ out of $2 x (x) + 2 x (- 5)$ .

$\frac{(x - 4) (x + 2)}{(x - 5) (x + 3)} \cdot \frac{2 x (x - 5)}{2 x^{2} - 8 x}$

Step 4.2

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

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Step 4.2.1

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

$\frac{(x - 4) (x + 2)}{(x - 5) (x + 3)} \cdot \frac{2 x (x - 5)}{2 x (x) - 8 x}$

Step 4.2.2

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

$\frac{(x - 4) (x + 2)}{(x - 5) (x + 3)} \cdot \frac{2 x (x - 5)}{2 x (x) + 2 x (- 4)}$

Step 4.2.3

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

$\frac{(x - 4) (x + 2)}{(x - 5) (x + 3)} \cdot \frac{2 x (x - 5)}{2 x (x - 4)}$

Step 4.3

Cancel the common factor of $x - 4$ .

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Step 4.3.1

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

$\frac{(x - 4) (x + 2)}{(x - 5) (x + 3)} \cdot \frac{2 x (x - 5)}{(x - 4) (2 x)}$

Step 4.3.2

Cancel the common factor.

$\frac{(x - 4) (x + 2)}{(x - 5) (x + 3)} \cdot \frac{2 x (x - 5)}{(x - 4) (2 x)}$

Step 4.3.3

Rewrite the expression.

$\frac{x + 2}{(x - 5) (x + 3)} \cdot \frac{2 x (x - 5)}{2 x}$

Step 4.4

Cancel the common factor of $x - 5$ .

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Step 4.4.1

Factor $x - 5$ out of $2 x (x - 5)$ .

$\frac{x + 2}{(x - 5) (x + 3)} \cdot \frac{(x - 5) (2 x)}{2 x}$

Step 4.4.2

Cancel the common factor.

$\frac{x + 2}{(x - 5) (x + 3)} \cdot \frac{(x - 5) (2 x)}{2 x}$

Step 4.4.3

Rewrite the expression.

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

Step 4.5

Multiply $\frac{x + 2}{x + 3}$ by $\frac{2 x}{2 x}$ .

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

Step 4.6

Cancel the common factor of $2$ .

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Step 4.6.1

Cancel the common factor.

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

Step 4.6.2

Rewrite the expression.

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

Step 4.7

Cancel the common factor of $x$ .

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Step 4.7.1

Cancel the common factor.

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

Step 4.7.2

Rewrite the expression.

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