Basic Math Examples

$\frac{z^{2} - z - 30}{z^{2} - 4 z - 45} \div \frac{z^{2} + 7 z + 12}{z^{2} - 5 z - 36}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{z^{2} - z - 30}{z^{2} - 4 z - 45} \cdot \frac{z^{2} - 5 z - 36}{z^{2} + 7 z + 12}$

Step 2

Factor $z^{2} - z - 30$ using the AC method.

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Step 2.1

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 $- 30$ and whose sum is $- 1$ .

$- 6, 5$

Step 2.2

Write the factored form using these integers.

$\frac{(z - 6) (z + 5)}{z^{2} - 4 z - 45} \cdot \frac{z^{2} - 5 z - 36}{z^{2} + 7 z + 12}$

Step 3

Factor $z^{2} - 4 z - 45$ using the AC method.

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Step 3.1

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 $- 45$ and whose sum is $- 4$ .

$- 9, 5$

Step 3.2

Write the factored form using these integers.

$\frac{(z - 6) (z + 5)}{(z - 9) (z + 5)} \cdot \frac{z^{2} - 5 z - 36}{z^{2} + 7 z + 12}$

Step 4

Cancel the common factor of $z + 5$ .

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Step 4.1

Cancel the common factor.

$\frac{(z - 6) (z + 5)}{(z - 9) (z + 5)} \cdot \frac{z^{2} - 5 z - 36}{z^{2} + 7 z + 12}$

Step 4.2

Rewrite the expression.

$\frac{z - 6}{z - 9} \cdot \frac{z^{2} - 5 z - 36}{z^{2} + 7 z + 12}$

Step 5

Factor $z^{2} - 5 z - 36$ using the AC method.

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Step 5.1

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 $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Step 5.2

Write the factored form using these integers.

$\frac{z - 6}{z - 9} \cdot \frac{(z - 9) (z + 4)}{z^{2} + 7 z + 12}$

Step 6

Factor $z^{2} + 7 z + 12$ using the AC method.

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Step 6.1

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 $12$ and whose sum is $7$ .

$3, 4$

Step 6.2

Write the factored form using these integers.

$\frac{z - 6}{z - 9} \cdot \frac{(z - 9) (z + 4)}{(z + 3) (z + 4)}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $z - 9$ .

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Step 7.1.1

Cancel the common factor.

$\frac{z - 6}{z - 9} \cdot \frac{(z - 9) (z + 4)}{(z + 3) (z + 4)}$

Step 7.1.2

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $z + 4$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

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

$\frac{z - 6}{z + 3}$