Basic Math Examples

$\frac{4 z}{z^{2} + 2 z - 3} + \frac{7}{z - 1}$

Step 1

Factor $z^{2} + 2 z - 3$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 1.2

Write the factored form using these integers.

$\frac{4 z}{(z - 1) (z + 3)} + \frac{7}{z - 1}$

Step 2

To write $\frac{7}{z - 1}$ as a fraction with a common denominator, multiply by $\frac{z + 3}{z + 3}$ .

$\frac{4 z}{(z - 1) (z + 3)} + \frac{7}{z - 1} \cdot \frac{z + 3}{z + 3}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{7}{z - 1}$ by $\frac{z + 3}{z + 3}$ .

$\frac{4 z}{(z - 1) (z + 3)} + \frac{7 (z + 3)}{(z - 1) (z + 3)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{4 z + 7 (z + 3)}{(z - 1) (z + 3)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{4 z + 7 z + 7 \cdot 3}{(z - 1) (z + 3)}$

Step 4.2

Multiply $7$ by $3$ .

$\frac{4 z + 7 z + 21}{(z - 1) (z + 3)}$

Step 4.3

Add $4 z$ and $7 z$ .

$\frac{11 z + 21}{(z - 1) (z + 3)}$