Finite Math Examples

$\frac{z}{z^{2} - 4} + \frac{6 z}{z^{2} - 2 z}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 2$ .

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

Step 1.2

Factor $z$ out of $z^{2} - 2 z$ .

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Step 1.2.1

Factor $z$ out of $z^{2}$ .

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

Step 1.2.2

Factor $z$ out of $- 2 z$ .

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

Step 1.2.3

Factor $z$ out of $z \cdot z + z \cdot - 2$ .

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

Step 1.3

Cancel the common factor of $z$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Reorder the factors of $(z - 2) (z + 2)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $6$ by $2$ .

$\frac{z + 6 z + 12}{(z + 2) (z - 2)}$

Step 5.3

Add $z$ and $6 z$ .

$\frac{7 z + 12}{(z + 2) (z - 2)}$