Basic Math Examples

$\frac{z^{2} + 4 z - 12}{z^{2} + 2 z - 8} \cdot \frac{z^{2} - 2 z - 8}{z^{2} + 8 z + 12}$

Step 1

Factor $z^{2} + 4 z - 12$ 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 $- 12$ and whose sum is $4$ .

$- 2, 6$

Step 1.2

Write the factored form using these integers.

$\frac{(z - 2) (z + 6)}{z^{2} + 2 z - 8} \cdot \frac{z^{2} - 2 z - 8}{z^{2} + 8 z + 12}$

Step 2

Factor $z^{2} + 2 z - 8$ 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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 2.2

Write the factored form using these integers.

$\frac{(z - 2) (z + 6)}{(z - 2) (z + 4)} \cdot \frac{z^{2} - 2 z - 8}{z^{2} + 8 z + 12}$

Step 3

Factor $z^{2} - 2 z - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 3.2

Write the factored form using these integers.

$\frac{(z - 2) (z + 6)}{(z - 2) (z + 4)} \cdot \frac{(z - 4) (z + 2)}{z^{2} + 8 z + 12}$

Step 4

Factor $z^{2} + 8 z + 12$ 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 $8$ .

$2, 6$

Step 4.2

Write the factored form using these integers.

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $z + 6$ .

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Step 5.1.1

Factor $z + 6$ out of $(z - 2) (z + 6)$ .

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

Step 5.1.2

Factor $z + 6$ out of $(z + 2) (z + 6)$ .

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

Step 5.1.3

Cancel the common factor.

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

Step 5.1.4

Rewrite the expression.

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

Step 5.2

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

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

Step 5.3

Cancel the common factor of $z - 2$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

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

Step 5.4

Cancel the common factor of $z + 2$ .

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Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Rewrite the expression.

$\frac{z - 4}{z + 4}$