Algebra Examples

$\frac{4}{b - 4} + \frac{3}{3 b^{2} - 18 b}$

Step 1

Simplify each term.

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Step 1.1

Factor $3 b$ out of $3 b^{2} - 18 b$ .

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Step 1.1.1

Factor $3 b$ out of $3 b^{2}$ .

$\frac{4}{b - 4} + \frac{3}{3 b (b) - 18 b}$

Step 1.1.2

Factor $3 b$ out of $- 18 b$ .

$\frac{4}{b - 4} + \frac{3}{3 b (b) + 3 b (- 6)}$

Step 1.1.3

Factor $3 b$ out of $3 b (b) + 3 b (- 6)$ .

$\frac{4}{b - 4} + \frac{3}{3 b (b - 6)}$

Step 1.2

Cancel the common factor of $3$ .

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Step 1.2.1

Cancel the common factor.

$\frac{4}{b - 4} + \frac{3}{3 b (b - 6)}$

Step 1.2.2

Rewrite the expression.

$\frac{4}{b - 4} + \frac{1}{b (b - 6)}$

Step 2

To write $\frac{4}{b - 4}$ as a fraction with a common denominator, multiply by $\frac{b (b - 6)}{b (b - 6)}$ .

$\frac{4}{b - 4} \cdot \frac{b (b - 6)}{b (b - 6)} + \frac{1}{b (b - 6)}$

Step 3

To write $\frac{1}{b (b - 6)}$ as a fraction with a common denominator, multiply by $\frac{b - 4}{b - 4}$ .

$\frac{4}{b - 4} \cdot \frac{b (b - 6)}{b (b - 6)} + \frac{1}{b (b - 6)} \cdot \frac{b - 4}{b - 4}$

Step 4

Write each expression with a common denominator of $(b - 4) b (b - 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{4}{b - 4}$ by $\frac{b (b - 6)}{b (b - 6)}$ .

$\frac{4 (b (b - 6))}{(b - 4) (b (b - 6))} + \frac{1}{b (b - 6)} \cdot \frac{b - 4}{b - 4}$

Step 4.2

Multiply $\frac{1}{b (b - 6)}$ by $\frac{b - 4}{b - 4}$ .

$\frac{4 (b (b - 6))}{(b - 4) (b (b - 6))} + \frac{b - 4}{b (b - 6) (b - 4)}$

Step 4.3

Reorder the factors of $(b - 4) (b (b - 6))$ .

$\frac{4 (b (b - 6))}{b (b - 4) (b - 6)} + \frac{b - 4}{b (b - 6) (b - 4)}$

Step 4.4

Reorder the factors of $b (b - 6) (b - 4)$ .

$\frac{4 (b (b - 6))}{b (b - 4) (b - 6)} + \frac{b - 4}{b (b - 4) (b - 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{4 (b (b - 6)) + b - 4}{b (b - 4) (b - 6)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{4 b \cdot b + 4 b \cdot - 6 + b - 4}{b (b - 4) (b - 6)}$

Step 6.2

Multiply $b$ by $b$ by adding the exponents.

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Step 6.2.1

Move $b$ .

$\frac{4 (b \cdot b) + 4 b \cdot - 6 + b - 4}{b (b - 4) (b - 6)}$

Step 6.2.2

Multiply $b$ by $b$ .

$\frac{4 b^{2} + 4 b \cdot - 6 + b - 4}{b (b - 4) (b - 6)}$

Step 6.3

Multiply $- 6$ by $4$ .

$\frac{4 b^{2} - 24 b + b - 4}{b (b - 4) (b - 6)}$

Step 6.4

Add $- 24 b$ and $b$ .

$\frac{4 b^{2} - 23 b - 4}{b (b - 4) (b - 6)}$