Algebra Examples

$\frac{x^{2} - 4 x - 5}{x - 2} \div \frac{2 x - 10}{x^{2} - 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 4 x - 5}{x - 2} \cdot \frac{x^{2} - 4}{2 x - 10}$

Step 2

Factor $x^{2} - 4 x - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 1)}{x - 2} \cdot \frac{x^{2} - 4}{2 x - 10}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{(x - 5) (x + 1)}{x - 2} \cdot \frac{x^{2} - 2^{2}}{2 x - 10}$

Step 3.2

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 $b = 2$ .

$\frac{(x - 5) (x + 1)}{x - 2} \cdot \frac{(x + 2) (x - 2)}{2 x - 10}$

Step 4

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

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Step 4.1

Factor $2$ out of $2 x$ .

$\frac{(x - 5) (x + 1)}{x - 2} \cdot \frac{(x + 2) (x - 2)}{2 (x) - 10}$

Step 4.2

Factor $2$ out of $- 10$ .

$\frac{(x - 5) (x + 1)}{x - 2} \cdot \frac{(x + 2) (x - 2)}{2 x + 2 \cdot - 5}$

Step 4.3

Factor $2$ out of $2 x + 2 \cdot - 5$ .

$\frac{(x - 5) (x + 1)}{x - 2} \cdot \frac{(x + 2) (x - 2)}{2 (x - 5)}$

Step 5

Cancel the common factor of $x - 5$ .

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Step 5.1

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

$\frac{(x - 5) (x + 1)}{x - 2} \cdot \frac{(x + 2) (x - 2)}{(x - 5) \cdot 2}$

Step 5.2

Cancel the common factor.

$\frac{(x - 5) (x + 1)}{x - 2} \cdot \frac{(x + 2) (x - 2)}{(x - 5) \cdot 2}$

Step 5.3

Rewrite the expression.

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

Step 6

Cancel the common factor of $x - 2$ .

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Apply the distributive property.

$x \frac{x + 2}{2} + 1 \frac{x + 2}{2}$

Step 8

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

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

Step 9

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify each term.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Multiply $x$ by $x$ .

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

Step 11.3

Move $2$ to the left of $x$ .

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

Step 12

Add $2 x$ and $x$ .

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

Step 13

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

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Step 13.1

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 $2$ and whose sum is $3$ .

$1, 2$

Step 13.2

Write the factored form using these integers.

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

Step 14

Expand $(x + 1) (x + 2)$ using the FOIL Method.

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Step 14.1

Apply the distributive property.

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

Step 14.2

Apply the distributive property.

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

Step 14.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 + 1 x + 1 \cdot 2}{2}$

Step 15

Simplify and combine like terms.

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Step 15.1

Simplify each term.

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Step 15.1.1

Multiply $x$ by $x$ .

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

Step 15.1.2

Move $2$ to the left of $x$ .

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

Step 15.1.3

Multiply $x$ by $1$ .

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

Step 15.1.4

Multiply $2$ by $1$ .

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

Step 15.2

Add $2 x$ and $x$ .

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

Step 16

Split the fraction $\frac{x^{2} + 3 x + 2}{2}$ into two fractions.

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

Step 17

Split the fraction $\frac{x^{2} + 3 x}{2}$ into two fractions.

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

Step 18

Divide $2$ by $2$ .

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