Algebra Examples

$\frac{x}{x^{2} - 25} + \frac{5}{x + 5} - \frac{6}{x}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{x}{x^{2} - 5^{2}} + \frac{5}{x + 5} - \frac{6}{x}$

Step 1.2

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 $b = 5$ .

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

Step 2

Find the common denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $\frac{6}{x}$ by $\frac{(x + 5) (x - 5)}{(x + 5) (x - 5)}$ .

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

Step 2.6

Multiply $\frac{6}{x}$ by $\frac{(x + 5) (x - 5)}{(x + 5) (x - 5)}$ .

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

Step 2.7

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

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

Step 2.8

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Multiply $x$ by $x$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Multiply $x$ by $x$ .

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Multiply $- 5$ by $5$ .

$\frac{x^{2} + 5 x^{2} - 25 x - 6 ((x + 5) (x - 5))}{x (x + 5) (x - 5)}$

Step 4.6

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

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Step 4.6.1

Apply the distributive property.

$\frac{x^{2} + 5 x^{2} - 25 x - 6 (x (x - 5) + 5 (x - 5))}{x (x + 5) (x - 5)}$

Step 4.6.2

Apply the distributive property.

$\frac{x^{2} + 5 x^{2} - 25 x - 6 (x \cdot x + x \cdot - 5 + 5 (x - 5))}{x (x + 5) (x - 5)}$

Step 4.6.3

Apply the distributive property.

$\frac{x^{2} + 5 x^{2} - 25 x - 6 (x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5)}{x (x + 5) (x - 5)}$

Step 4.7

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

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Step 4.7.1

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

$\frac{x^{2} + 5 x^{2} - 25 x - 6 (x \cdot x - 5 x + 5 x + 5 \cdot - 5)}{x (x + 5) (x - 5)}$

Step 4.7.2

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

$\frac{x^{2} + 5 x^{2} - 25 x - 6 (x \cdot x + 0 + 5 \cdot - 5)}{x (x + 5) (x - 5)}$

Step 4.7.3

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

$\frac{x^{2} + 5 x^{2} - 25 x - 6 (x \cdot x + 5 \cdot - 5)}{x (x + 5) (x - 5)}$

Step 4.8

Simplify each term.

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Step 4.8.1

Multiply $x$ by $x$ .

$\frac{x^{2} + 5 x^{2} - 25 x - 6 (x^{2} + 5 \cdot - 5)}{x (x + 5) (x - 5)}$

Step 4.8.2

Multiply $5$ by $- 5$ .

$\frac{x^{2} + 5 x^{2} - 25 x - 6 (x^{2} - 25)}{x (x + 5) (x - 5)}$

Step 4.9

Apply the distributive property.

$\frac{x^{2} + 5 x^{2} - 25 x - 6 x^{2} - 6 \cdot - 25}{x (x + 5) (x - 5)}$

Step 4.10

Multiply $- 6$ by $- 25$ .

$\frac{x^{2} + 5 x^{2} - 25 x - 6 x^{2} + 150}{x (x + 5) (x - 5)}$

Step 5

Simplify terms.

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Step 5.1

Add $x^{2}$ and $5 x^{2}$ .

$\frac{6 x^{2} - 25 x - 6 x^{2} + 150}{x (x + 5) (x - 5)}$

Step 5.2

Combine the opposite terms in $6 x^{2} - 25 x - 6 x^{2} + 150$ .

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Step 5.2.1

Subtract $6 x^{2}$ from $6 x^{2}$ .

$\frac{- 25 x + 0 + 150}{x (x + 5) (x - 5)}$

Step 5.2.2

Add $- 25 x$ and $0$ .

$\frac{- 25 x + 150}{x (x + 5) (x - 5)}$

Step 5.3

Factor $25$ out of $- 25 x + 150$ .

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Step 5.3.1

Factor $25$ out of $- 25 x$ .

$\frac{25 (- x) + 150}{x (x + 5) (x - 5)}$

Step 5.3.2

Factor $25$ out of $150$ .

$\frac{25 (- x) + 25 (6)}{x (x + 5) (x - 5)}$

Step 5.3.3

Factor $25$ out of $25 (- x) + 25 (6)$ .

$\frac{25 (- x + 6)}{x (x + 5) (x - 5)}$

Step 5.4

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

$\frac{25 (- (x) + 6)}{x (x + 5) (x - 5)}$

Step 5.5

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{25 (- (x) - 1 (- 6))}{x (x + 5) (x - 5)}$

Step 5.6

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

$\frac{25 (- (x - 6))}{x (x + 5) (x - 5)}$

Step 5.7

Simplify the expression.

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Step 5.7.1

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

$\frac{25 (- 1 (x - 6))}{x (x + 5) (x - 5)}$

Step 5.7.2

Move the negative in front of the fraction.

$- \frac{25 (x - 6)}{x (x + 5) (x - 5)}$