Basic Math Examples

$\frac{z^{2} + z - 6}{5 z} \cdot \frac{5 z - 10}{z + 3}$

Step 1

Factor $z^{2} + z - 6$ 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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 1.2

Write the factored form using these integers.

$\frac{(z - 2) (z + 3)}{5 z} \cdot \frac{5 z - 10}{z + 3}$

Step 2

Simplify terms.

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Step 2.1

Factor $5$ out of $5 z - 10$ .

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Step 2.1.1

Factor $5$ out of $5 z$ .

$\frac{(z - 2) (z + 3)}{5 z} \cdot \frac{5 (z) - 10}{z + 3}$

Step 2.1.2

Factor $5$ out of $- 10$ .

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

Step 2.1.3

Factor $5$ out of $5 z + 5 \cdot - 2$ .

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

Step 2.2

Cancel the common factor of $z + 3$ .

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Step 2.2.1

Factor $z + 3$ out of $(z - 2) (z + 3)$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 2.3

Cancel the common factor of $5$ .

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Step 2.3.1

Factor $5$ out of $5 z$ .

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

Step 2.3.2

Cancel the common factor.

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

Step 2.3.3

Rewrite the expression.

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

Step 2.4

Multiply $\frac{z - 2}{z}$ by $z - 2$ .

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

Step 3

Simplify the numerator.

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Step 3.1

Raise $z - 2$ to the power of $1$ .

$\frac{{(z - 2)}^{1} (z - 2)}{z}$

Step 3.2

Raise $z - 2$ to the power of $1$ .

$\frac{{(z - 2)}^{1} {(z - 2)}^{1}}{z}$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(z - 2)}^{1 + 1}}{z}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(z - 2)}^{2}}{z}$