Basic Math Examples

$\frac{z^{2} - 4}{z - 3} \div \frac{z + 2}{z^{2} + z - 12}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{z^{2} - 4}{z - 3} \cdot \frac{z^{2} + z - 12}{z + 2}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{z^{2} - 2^{2}}{z - 3} \cdot \frac{z^{2} + z - 12}{z + 2}$

Step 2.2

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 $b = 2$ .

$\frac{(z + 2) (z - 2)}{z - 3} \cdot \frac{z^{2} + z - 12}{z + 2}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $z + 2$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(z + 2) (z - 2)}{z - 3} \cdot \frac{z^{2} + z - 12}{z + 2}$

Step 3.1.2

Rewrite the expression.

$\frac{z - 2}{z - 3} (z^{2} + z - 12)$

Step 3.2

Multiply $\frac{z - 2}{z - 3}$ by $z^{2} + z - 12$ .

$\frac{(z - 2) (z^{2} + z - 12)}{z - 3}$

Step 4

Factor $z^{2} + z - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 4.2

Write the factored form using these integers.

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

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

Step 5

Cancel the common factor of $z - 3$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $(z - 2) (z + 4)$ by $1$ .

$(z - 2) (z + 4)$

Step 6

Expand $(z - 2) (z + 4)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$z (z + 4) - 2 (z + 4)$

Step 6.2

Apply the distributive property.

$z \cdot z + z \cdot 4 - 2 (z + 4)$

Step 6.3

Apply the distributive property.

$z \cdot z + z \cdot 4 - 2 z - 2 \cdot 4$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $z$ by $z$ .

$z^{2} + z \cdot 4 - 2 z - 2 \cdot 4$

Step 7.1.2

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

$z^{2} + 4 \cdot z - 2 z - 2 \cdot 4$

Step 7.1.3

Multiply $- 2$ by $4$ .

$z^{2} + 4 z - 2 z - 8$

Step 7.2

Subtract $2 z$ from $4 z$ .

$z^{2} + 2 z - 8$