Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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

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

Step 2

Find the common denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Multiply $b$ by $b$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Multiply $b$ by $b$ .

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Multiply $- 5$ by $5$ .

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

Step 4.6

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

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Step 4.6.1

Apply the distributive property.

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

Step 4.6.2

Apply the distributive property.

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

Step 4.6.3

Apply the distributive property.

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

Add $- 5 b$ and $5 b$ .

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

Step 4.7.3

Add $b \cdot b$ and $0$ .

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

Step 4.8

Simplify each term.

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Step 4.8.1

Multiply $b$ by $b$ .

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

Step 4.8.2

Multiply $5$ by $- 5$ .

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

Step 4.9

Apply the distributive property.

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

Step 4.10

Multiply $- 6$ by $- 25$ .

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

Step 5

Simplify terms.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Add $- 25 b$ and $0$ .

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

Step 5.3

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

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Step 5.3.1

Factor $25$ out of $- 25 b$ .

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

Step 5.3.2

Factor $25$ out of $150$ .

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

Step 5.3.3

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

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

Step 5.4

Factor $- 1$ out of $- b$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

Simplify the expression.

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Step 5.7.1

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

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

Step 5.7.2

Move the negative in front of the fraction.

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