Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $2$ out of $2 x$ .

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

Step 1.1.2

Factor $2$ out of $10$ .

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

Step 1.1.3

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

$\frac{x + 4}{2 (x + 5)} - \frac{5}{x^{2} - 25}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

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 $b = 5$ .

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

Step 2

To write $\frac{x + 4}{2 (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

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

Step 3

To write $- \frac{5}{(x + 5) (x - 5)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4

Write each expression with a common denominator of $2 (x + 5) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x + 4}{2 (x + 5)}$ by $\frac{x - 5}{x - 5}$ .

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x + 5) (x - 5) \cdot 2$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Expand $(x + 4) (x - 5)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

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

Step 6.2.1.2

Move $- 5$ to the left of $x$ .

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

Step 6.2.1.3

Multiply $4$ by $- 5$ .

$\frac{x^{2} - 5 x + 4 x - 20 - 5 \cdot 2}{2 (x + 5) (x - 5)}$

Step 6.2.2

Add $- 5 x$ and $4 x$ .

$\frac{x^{2} - x - 20 - 5 \cdot 2}{2 (x + 5) (x - 5)}$

Step 6.3

Multiply $- 5$ by $2$ .

$\frac{x^{2} - x - 20 - 10}{2 (x + 5) (x - 5)}$

Step 6.4

Subtract $10$ from $- 20$ .

$\frac{x^{2} - x - 30}{2 (x + 5) (x - 5)}$

Step 6.5

Factor $x^{2} - x - 30$ using the AC method.

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Step 6.5.1

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 $- 30$ and whose sum is $- 1$ .

$- 6, 5$

Step 6.5.2

Write the factored form using these integers.

$\frac{(x - 6) (x + 5)}{2 (x + 5) (x - 5)}$

Step 7

Cancel the common factor of $x + 5$ .

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Step 7.1

Cancel the common factor.

$\frac{(x - 6) (x + 5)}{2 (x + 5) (x - 5)}$

Step 7.2

Rewrite the expression.

$\frac{x - 6}{2 (x - 5)}$