Algebra Examples

$\frac{\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}}{\frac{14 x^{2} - 22 x - 12}{x^{2} - 25}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{x^{2} - 25}{14 x^{2} - 22 x - 12}$

Step 2

Simplify the numerator.

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Step 2.1

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

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{x^{2} - 5^{2}}{14 x^{2} - 22 x - 12}$

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 $a = x$ and $b = 5$ .

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{14 x^{2} - 22 x - 12}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $2$ out of $14 x^{2} - 22 x - 12$ .

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Step 3.1.1

Factor $2$ out of $14 x^{2}$ .

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x^{2}) - 22 x - 12}$

Step 3.1.2

Factor $2$ out of $- 22 x$ .

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x^{2}) + 2 (- 11 x) - 12}$

Step 3.1.3

Factor $2$ out of $- 12$ .

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x^{2}) + 2 (- 11 x) + 2 (- 6)}$

Step 3.1.4

Factor $2$ out of $2 (7 x^{2}) + 2 (- 11 x)$ .

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x^{2} - 11 x) + 2 (- 6)}$

Step 3.1.5

Factor $2$ out of $2 (7 x^{2} - 11 x) + 2 (- 6)$ .

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x^{2} - 11 x - 6)}$

Step 3.2

Factor by grouping.

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Step 3.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 7 \cdot - 6 = - 42$ and whose sum is $b = - 11$ .

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Step 3.2.1.1

Factor $- 11$ out of $- 11 x$ .

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x^{2} - 11 (x) - 6)}$

Step 3.2.1.2

Rewrite $- 11$ as $3$ plus $- 14$

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x^{2} + (3 - 14) x - 6)}$

Step 3.2.1.3

Apply the distributive property.

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x^{2} + 3 x - 14 x - 6)}$

Step 3.2.2

Factor out the greatest common factor from each group.

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Step 3.2.2.1

Group the first two terms and the last two terms.

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 ((7 x^{2} + 3 x) - 14 x - 6)}$

Step 3.2.2.2

Factor out the greatest common factor (GCF) from each group.

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (x (7 x + 3) - 2 (7 x + 3))}$

Step 3.2.3

Factor the polynomial by factoring out the greatest common factor, $7 x + 3$ .

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 ((7 x + 3) (x - 2))}$

$(\frac{3}{x^{2} + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 4

Simplify each term.

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Step 4.1

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

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Step 4.1.1

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

$(\frac{3}{x \cdot x + 5 x} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 4.1.2

Factor $x$ out of $5 x$ .

$(\frac{3}{x \cdot x + x \cdot 5} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 4.1.3

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

$(\frac{3}{x (x + 5)} + \frac{4}{x^{2} + 6 x + 5}) \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 4.2

Factor $x^{2} + 6 x + 5$ using the AC method.

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Step 4.2.1

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 $6$ .

$1, 5$

Step 4.2.2

Write the factored form using these integers.

$(\frac{3}{x (x + 5)} + \frac{4}{(x + 1) (x + 5)}) \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 5

To write $\frac{3}{x (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$(\frac{3}{x (x + 5)} \cdot \frac{x + 1}{x + 1} + \frac{4}{(x + 1) (x + 5)}) \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 6

To write $\frac{4}{(x + 1) (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$(\frac{3}{x (x + 5)} \cdot \frac{x + 1}{x + 1} + \frac{4}{(x + 1) (x + 5)} \cdot \frac{x}{x}) \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 7

Write each expression with a common denominator of $x (x + 5) (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{3}{x (x + 5)}$ by $\frac{x + 1}{x + 1}$ .

$(\frac{3 (x + 1)}{x (x + 5) (x + 1)} + \frac{4}{(x + 1) (x + 5)} \cdot \frac{x}{x}) \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 7.2

Multiply $\frac{4}{(x + 1) (x + 5)}$ by $\frac{x}{x}$ .

$(\frac{3 (x + 1)}{x (x + 5) (x + 1)} + \frac{4 x}{(x + 1) (x + 5) x}) \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 7.3

Reorder the factors of $x (x + 5) (x + 1)$ .

$(\frac{3 (x + 1)}{x (x + 1) (x + 5)} + \frac{4 x}{(x + 1) (x + 5) x}) \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 7.4

Reorder the factors of $(x + 1) (x + 5) x$ .

$(\frac{3 (x + 1)}{x (x + 1) (x + 5)} + \frac{4 x}{x (x + 1) (x + 5)}) \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 8

Combine the numerators over the common denominator.

$\frac{3 (x + 1) + 4 x}{x (x + 1) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

$\frac{3 x + 3 \cdot 1 + 4 x}{x (x + 1) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 9.2

Multiply $3$ by $1$ .

$\frac{3 x + 3 + 4 x}{x (x + 1) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 9.3

Add $3 x$ and $4 x$ .

$\frac{7 x + 3}{x (x + 1) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{2 (7 x + 3) (x - 2)}$

Step 10

Simplify terms.

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Step 10.1

Cancel the common factor of $7 x + 3$ .

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Step 10.1.1

Factor $7 x + 3$ out of $2 (7 x + 3) (x - 2)$ .

$\frac{7 x + 3}{x (x + 1) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{(7 x + 3) (2 (x - 2))}$

Step 10.1.2

Cancel the common factor.

$\frac{7 x + 3}{x (x + 1) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{(7 x + 3) (2 (x - 2))}$

Step 10.1.3

Rewrite the expression.

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

Step 10.2

Cancel the common factor of $x + 5$ .

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Step 10.2.1

Factor $x + 5$ out of $x (x + 1) (x + 5)$ .

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

Step 10.2.2

Cancel the common factor.

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

Step 10.2.3

Rewrite the expression.

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

Step 10.3

Multiply $\frac{1}{x (x + 1)}$ by $\frac{x - 5}{2 (x - 2)}$ .

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

Step 10.4

Move $2$ to the left of $x (x + 1)$ .

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