Basic Math Examples

$\frac{z^{2} - 3 z - 18}{24 - 4 z} \div \frac{z^{2} - 2 z - 15}{3 z + 6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{z^{2} - 3 z - 18}{24 - 4 z} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 2

Factor $z^{2} - 3 z - 18$ 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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(z - 6) (z + 3)}{24 - 4 z} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 3

Simplify terms.

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Step 3.1

Factor $4$ out of $24 - 4 z$ .

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Step 3.1.1

Factor $4$ out of $24$ .

$\frac{(z - 6) (z + 3)}{4 (6) - 4 z} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 3.1.2

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

$\frac{(z - 6) (z + 3)}{4 (6) + 4 (- z)} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 3.1.3

Factor $4$ out of $4 (6) + 4 (- z)$ .

$\frac{(z - 6) (z + 3)}{4 (6 - z)} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 3.2

Cancel the common factor of $z - 6$ and $6 - z$ .

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Step 3.2.1

Factor $- 1$ out of $z$ .

$\frac{(- 1 (- z) - 6) (z + 3)}{4 (6 - z)} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 3.2.2

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{(- 1 (- z) - 1 (6)) (z + 3)}{4 (6 - z)} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 3.2.3

Factor $- 1$ out of $- 1 (- z) - 1 (6)$ .

$\frac{- 1 (- z + 6) (z + 3)}{4 (6 - z)} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 3.2.4

Reorder terms.

$\frac{- 1 (- z + 6) (z + 3)}{4 (- z + 6)} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 3.2.5

Cancel the common factor.

$\frac{- 1 (- z + 6) (z + 3)}{4 (- z + 6)} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 3.2.6

Rewrite the expression.

$\frac{- 1 (z + 3)}{4} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 3.3

Move the negative in front of the fraction.

$- \frac{z + 3}{4} \cdot \frac{3 z + 6}{z^{2} - 2 z - 15}$

Step 3.4

Factor $3$ out of $3 z + 6$ .

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Step 3.4.1

Factor $3$ out of $3 z$ .

$- \frac{z + 3}{4} \cdot \frac{3 (z) + 6}{z^{2} - 2 z - 15}$

Step 3.4.2

Factor $3$ out of $6$ .

$- \frac{z + 3}{4} \cdot \frac{3 z + 3 \cdot 2}{z^{2} - 2 z - 15}$

Step 3.4.3

Factor $3$ out of $3 z + 3 \cdot 2$ .

$- \frac{z + 3}{4} \cdot \frac{3 (z + 2)}{z^{2} - 2 z - 15}$

Step 4

Factor $z^{2} - 2 z - 15$ using the AC method.

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Step 4.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 4.2

Write the factored form using these integers.

$- \frac{z + 3}{4} \cdot \frac{3 (z + 2)}{(z - 5) (z + 3)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $z + 3$ .

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Step 5.1.1

Move the leading negative in $- \frac{z + 3}{4}$ into the numerator.

$\frac{- (z + 3)}{4} \cdot \frac{3 (z + 2)}{(z - 5) (z + 3)}$

Step 5.1.2

Factor $z + 3$ out of $- (z + 3)$ .

$\frac{(z + 3) \cdot - 1}{4} \cdot \frac{3 (z + 2)}{(z - 5) (z + 3)}$

Step 5.1.3

Factor $z + 3$ out of $(z - 5) (z + 3)$ .

$\frac{(z + 3) \cdot - 1}{4} \cdot \frac{3 (z + 2)}{(z + 3) (z - 5)}$

Step 5.1.4

Cancel the common factor.

$\frac{(z + 3) \cdot - 1}{4} \cdot \frac{3 (z + 2)}{(z + 3) (z - 5)}$

Step 5.1.5

Rewrite the expression.

$\frac{- 1}{4} \cdot \frac{3 (z + 2)}{z - 5}$

Step 5.2

Multiply $\frac{- 1}{4}$ by $\frac{3 (z + 2)}{z - 5}$ .

$\frac{- (3 (z + 2))}{4 (z - 5)}$

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $3$ by $- 1$ .

$\frac{- 3 (z + 2)}{4 (z - 5)}$

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{3 (z + 2)}{4 (z - 5)}$