Algebra Examples

$\frac{4 - x^{2}}{x - 2} \cdot \frac{x^{2} - 25}{x + 2} \div \frac{x^{2} + 10 x + 25}{x + 5}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 - x^{2}}{x - 2} \cdot \frac{x^{2} - 25}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{2^{2} - x^{2}}{x - 2} \cdot \frac{x^{2} - 25}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x$ .

$\frac{(2 + x) (2 - x)}{x - 2} \cdot \frac{x^{2} - 25}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $25$ as $5^{2}$ .

$\frac{(2 + x) (2 - x)}{x - 2} \cdot \frac{x^{2} - 5^{2}}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

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 $b = 5$ .

$\frac{(2 + x) (2 - x)}{x - 2} \cdot \frac{(x + 5) (x - 5)}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4

Simplify terms.

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Step 4.1

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

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Step 4.1.1

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{(2 + x) (- 1 (- 2) - x)}{x - 2} \cdot \frac{(x + 5) (x - 5)}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.1.2

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

$\frac{(2 + x) (- 1 (- 2) - (x))}{x - 2} \cdot \frac{(x + 5) (x - 5)}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.1.3

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

$\frac{(2 + x) (- 1 (- 2 + x))}{x - 2} \cdot \frac{(x + 5) (x - 5)}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.1.4

Reorder terms.

$\frac{(2 + x) (- 1 (x - 2))}{x - 2} \cdot \frac{(x + 5) (x - 5)}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.1.5

Cancel the common factor.

$\frac{(2 + x) (- 1 (x - 2))}{x - 2} \cdot \frac{(x + 5) (x - 5)}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.1.6

Divide $(2 + x) \cdot (- 1)$ by $1$ .

$(2 + x) \cdot (- 1) \cdot \frac{(x + 5) (x - 5)}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.2

Simplify by multiplying through.

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Step 4.2.1

Apply the distributive property.

$(2 \cdot - 1 + x \cdot - 1) \cdot \frac{(x + 5) (x - 5)}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Multiply $2$ by $- 1$ .

$(- 2 + x \cdot - 1) \cdot \frac{(x + 5) (x - 5)}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.2.2.2

Move $- 1$ to the left of $x$ .

$(- 2 - 1 \cdot x) \cdot \frac{(x + 5) (x - 5)}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.3

Rewrite $- 1 x$ as $- x$ .

$(- 2 - x) \cdot \frac{(x + 5) (x - 5)}{x + 2} \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.4

Simplify terms.

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Step 4.4.1

Multiply $- 2 - x$ by $\frac{(x + 5) (x - 5)}{x + 2}$ .

$\frac{(- 2 - x) ((x + 5) (x - 5))}{x + 2} \cdot \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.4.2

Cancel the common factor of $- 2 - x$ and $x + 2$ .

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Step 4.4.2.1

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{(- 1 (2) - x) ((x + 5) (x - 5))}{x + 2} \cdot \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.4.2.2

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

$\frac{(- 1 (2) - (x)) ((x + 5) (x - 5))}{x + 2} \cdot \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.4.2.3

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

$\frac{- 1 (2 + x) ((x + 5) (x - 5))}{x + 2} \cdot \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.4.2.4

Reorder terms.

$\frac{- 1 (x + 2) ((x + 5) (x - 5))}{x + 2} \cdot \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.4.2.5

Cancel the common factor.

$\frac{- 1 (x + 2) ((x + 5) (x - 5))}{x + 2} \cdot \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.4.2.6

Divide $- 1 ((x + 5) (x - 5))$ by $1$ .

$- 1 ((x + 5) (x - 5)) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 4.4.3

Rewrite $- 1 ((x + 5) (x - 5))$ as $- ((x + 5) (x - 5))$ .

$- ((x + 5) (x - 5)) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 5

Expand $(x + 5) (x - 5)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$- (x (x - 5) + 5 (x - 5)) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 5.2

Apply the distributive property.

$- (x \cdot x + x \cdot - 5 + 5 (x - 5)) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 5.3

Apply the distributive property.

$- (x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 6

Simplify terms.

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Step 6.1

Combine the opposite terms in $x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5$ .

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Step 6.1.1

Reorder the factors in the terms $x \cdot - 5$ and $5 x$ .

$- (x \cdot x - 5 x + 5 x + 5 \cdot - 5) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 6.1.2

Add $- 5 x$ and $5 x$ .

$- (x \cdot x + 0 + 5 \cdot - 5) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 6.1.3

Add $x \cdot x$ and $0$ .

$- (x \cdot x + 5 \cdot - 5) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $x$ by $x$ .

$- (x^{2} + 5 \cdot - 5) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 6.2.2

Multiply $5$ by $- 5$ .

$- (x^{2} - 25) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 6.3

Simplify by multiplying through.

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Step 6.3.1

Apply the distributive property.

$(- x^{2} - - 25) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 6.3.2

Multiply $- 1$ by $- 25$ .

$(- x^{2} + 25) \frac{x + 5}{x^{2} + 10 x + 25}$

Step 7

Factor using the perfect square rule.

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Step 7.1

Rewrite $25$ as $5^{2}$ .

$(- x^{2} + 25) \frac{x + 5}{x^{2} + 10 x + 5^{2}}$

Step 7.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 7.3

Rewrite the polynomial.

$(- x^{2} + 25) \frac{x + 5}{x^{2} + 2 \cdot x \cdot 5 + 5^{2}}$

Step 7.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$(- x^{2} + 25) \frac{x + 5}{{(x + 5)}^{2}}$

Step 8

Simplify terms.

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Step 8.1

Cancel the common factor of $x + 5$ and ${(x + 5)}^{2}$ .

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Step 8.1.1

Multiply by $1$ .

$(- x^{2} + 25) \frac{(x + 5) \cdot 1}{{(x + 5)}^{2}}$

Step 8.1.2

Cancel the common factors.

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Step 8.1.2.1

Factor $x + 5$ out of ${(x + 5)}^{2}$ .

$(- x^{2} + 25) \frac{(x + 5) \cdot 1}{(x + 5) (x + 5)}$

Step 8.1.2.2

Cancel the common factor.

$(- x^{2} + 25) \frac{(x + 5) \cdot 1}{(x + 5) (x + 5)}$

Step 8.1.2.3

Rewrite the expression.

$(- x^{2} + 25) \frac{1}{x + 5}$

Step 8.2

Multiply $- x^{2} + 25$ by $\frac{1}{x + 5}$ .

$\frac{- x^{2} + 25}{x + 5}$

Step 9

Simplify the numerator.

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Step 9.1

Rewrite $25$ as $5^{2}$ .

$\frac{- x^{2} + 5^{2}}{x + 5}$

Step 9.2

Reorder $- x^{2}$ and $5^{2}$ .

$\frac{5^{2} - x^{2}}{x + 5}$

Step 9.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$\frac{(5 + x) (5 - x)}{x + 5}$

Step 10

Cancel the common factor of $5 + x$ and $x + 5$ .

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Step 10.1

Reorder terms.

$\frac{(x + 5) (5 - x)}{x + 5}$

Step 10.2

Cancel the common factor.

$\frac{(x + 5) (5 - x)}{x + 5}$

Step 10.3

Divide $5 - x$ by $1$ .

$5 - x$