Basic Math Examples

$\frac{z^{2} + 15 z + 54}{z^{2} + 14 z + 45} \div \frac{z^{2} - z - 54}{4 z - 36}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{z^{2} + 15 z + 54}{z^{2} + 14 z + 45} \cdot \frac{4 z - 36}{z^{2} - z - 54}$

Step 2

Factor $z^{2} + 15 z + 54$ 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 $54$ and whose sum is $15$ .

$6, 9$

Step 2.2

Write the factored form using these integers.

$\frac{(z + 6) (z + 9)}{z^{2} + 14 z + 45} \cdot \frac{4 z - 36}{z^{2} - z - 54}$

Step 3

Factor $z^{2} + 14 z + 45$ 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 $14$ .

$5, 9$

Step 3.2

Write the factored form using these integers.

$\frac{(z + 6) (z + 9)}{(z + 5) (z + 9)} \cdot \frac{4 z - 36}{z^{2} - z - 54}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $z + 9$ .

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Step 4.1.1

Cancel the common factor.

$\frac{(z + 6) (z + 9)}{(z + 5) (z + 9)} \cdot \frac{4 z - 36}{z^{2} - z - 54}$

Step 4.1.2

Rewrite the expression.

$\frac{z + 6}{z + 5} \cdot \frac{4 z - 36}{z^{2} - z - 54}$

Step 4.2

Factor $4$ out of $4 z - 36$ .

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Step 4.2.1

Factor $4$ out of $4 z$ .

$\frac{z + 6}{z + 5} \cdot \frac{4 (z) - 36}{z^{2} - z - 54}$

Step 4.2.2

Factor $4$ out of $- 36$ .

$\frac{z + 6}{z + 5} \cdot \frac{4 z + 4 \cdot - 9}{z^{2} - z - 54}$

Step 4.2.3

Factor $4$ out of $4 z + 4 \cdot - 9$ .

$\frac{z + 6}{z + 5} \cdot \frac{4 (z - 9)}{z^{2} - z - 54}$

Step 4.3

Multiply $\frac{z + 6}{z + 5}$ by $\frac{4 (z - 9)}{z^{2} - z - 54}$ .

$\frac{(z + 6) (4 (z - 9))}{(z + 5) (z^{2} - z - 54)}$

Step 4.4

Move $4$ to the left of $z + 6$ .

$\frac{4 (z + 6) (z - 9)}{(z + 5) (z^{2} - z - 54)}$