Algebra Examples

$\frac{6 z^{2}}{3 z - 1} + \frac{2 z^{2}}{3 z + 1}$

Step 1

To write $\frac{6 z^{2}}{3 z - 1}$ as a fraction with a common denominator, multiply by $\frac{3 z + 1}{3 z + 1}$ .

$\frac{6 z^{2}}{3 z - 1} \cdot \frac{3 z + 1}{3 z + 1} + \frac{2 z^{2}}{3 z + 1}$

Step 2

To write $\frac{2 z^{2}}{3 z + 1}$ as a fraction with a common denominator, multiply by $\frac{3 z - 1}{3 z - 1}$ .

$\frac{6 z^{2}}{3 z - 1} \cdot \frac{3 z + 1}{3 z + 1} + \frac{2 z^{2}}{3 z + 1} \cdot \frac{3 z - 1}{3 z - 1}$

Step 3

Write each expression with a common denominator of $(3 z - 1) (3 z + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{6 z^{2}}{3 z - 1}$ by $\frac{3 z + 1}{3 z + 1}$ .

$\frac{6 z^{2} (3 z + 1)}{(3 z - 1) (3 z + 1)} + \frac{2 z^{2}}{3 z + 1} \cdot \frac{3 z - 1}{3 z - 1}$

Step 3.2

Multiply $\frac{2 z^{2}}{3 z + 1}$ by $\frac{3 z - 1}{3 z - 1}$ .

$\frac{6 z^{2} (3 z + 1)}{(3 z - 1) (3 z + 1)} + \frac{2 z^{2} (3 z - 1)}{(3 z + 1) (3 z - 1)}$

Step 3.3

Reorder the factors of $(3 z - 1) (3 z + 1)$ .

$\frac{6 z^{2} (3 z + 1)}{(3 z + 1) (3 z - 1)} + \frac{2 z^{2} (3 z - 1)}{(3 z + 1) (3 z - 1)}$

Step 4

Combine the numerators over the common denominator.

$\frac{6 z^{2} (3 z + 1) + 2 z^{2} (3 z - 1)}{(3 z + 1) (3 z - 1)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $2 z^{2}$ out of $6 z^{2} (3 z + 1) + 2 z^{2} (3 z - 1)$ .

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Step 5.1.1

Factor $2 z^{2}$ out of $6 z^{2} (3 z + 1)$ .

$\frac{2 z^{2} (3 (3 z + 1)) + 2 z^{2} (3 z - 1)}{(3 z + 1) (3 z - 1)}$

Step 5.1.2

Factor $2 z^{2}$ out of $2 z^{2} (3 (3 z + 1)) + 2 z^{2} (3 z - 1)$ .

$\frac{2 z^{2} (3 (3 z + 1) + 3 z - 1)}{(3 z + 1) (3 z - 1)}$

Step 5.2

Apply the distributive property.

$\frac{2 z^{2} (3 (3 z) + 3 \cdot 1 + 3 z - 1)}{(3 z + 1) (3 z - 1)}$

Step 5.3

Multiply $3$ by $3$ .

$\frac{2 z^{2} (9 z + 3 \cdot 1 + 3 z - 1)}{(3 z + 1) (3 z - 1)}$

Step 5.4

Multiply $3$ by $1$ .

$\frac{2 z^{2} (9 z + 3 + 3 z - 1)}{(3 z + 1) (3 z - 1)}$

Step 5.5

Add $9 z$ and $3 z$ .

$\frac{2 z^{2} (12 z + 3 - 1)}{(3 z + 1) (3 z - 1)}$

Step 5.6

Subtract $1$ from $3$ .

$\frac{2 z^{2} (12 z + 2)}{(3 z + 1) (3 z - 1)}$

Step 5.7

Factor $2$ out of $12 z + 2$ .

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Step 5.7.1

Factor $2$ out of $12 z$ .

$\frac{2 z^{2} (2 (6 z) + 2)}{(3 z + 1) (3 z - 1)}$

Step 5.7.2

Factor $2$ out of $2$ .

$\frac{2 z^{2} (2 (6 z) + 2 (1))}{(3 z + 1) (3 z - 1)}$

Step 5.7.3

Factor $2$ out of $2 (6 z) + 2 (1)$ .

$\frac{2 z^{2} (2 (6 z + 1))}{(3 z + 1) (3 z - 1)}$

$\frac{2 z^{2} \cdot 2 (6 z + 1)}{(3 z + 1) (3 z - 1)}$

Step 5.8

Multiply $2$ by $2$ .

$\frac{4 z^{2} (6 z + 1)}{(3 z + 1) (3 z - 1)}$