Finite Math Examples

$\frac{x^{2} + 3 x - 18}{x - 1} \cdot \frac{2 x^{2} - 2 x}{x^{3} - 3 x^{2}}$

Step 1

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

$- 3, 6$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 6)}{x - 1} \cdot \frac{2 x^{2} - 2 x}{x^{3} - 3 x^{2}}$

Step 2

Simplify terms.

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Step 2.1

Factor $2 x$ out of $2 x^{2} - 2 x$ .

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Step 2.1.1

Factor $2 x$ out of $2 x^{2}$ .

$\frac{(x - 3) (x + 6)}{x - 1} \cdot \frac{2 x (x) - 2 x}{x^{3} - 3 x^{2}}$

Step 2.1.2

Factor $2 x$ out of $- 2 x$ .

$\frac{(x - 3) (x + 6)}{x - 1} \cdot \frac{2 x (x) + 2 x (- 1)}{x^{3} - 3 x^{2}}$

Step 2.1.3

Factor $2 x$ out of $2 x (x) + 2 x (- 1)$ .

$\frac{(x - 3) (x + 6)}{x - 1} \cdot \frac{2 x (x - 1)}{x^{3} - 3 x^{2}}$

Step 2.2

Factor $x^{2}$ out of $x^{3} - 3 x^{2}$ .

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Step 2.2.1

Factor $x^{2}$ out of $x^{3}$ .

$\frac{(x - 3) (x + 6)}{x - 1} \cdot \frac{2 x (x - 1)}{x^{2} x - 3 x^{2}}$

Step 2.2.2

Factor $x^{2}$ out of $- 3 x^{2}$ .

$\frac{(x - 3) (x + 6)}{x - 1} \cdot \frac{2 x (x - 1)}{x^{2} x + x^{2} \cdot - 3}$

Step 2.2.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 3$ .

$\frac{(x - 3) (x + 6)}{x - 1} \cdot \frac{2 x (x - 1)}{x^{2} (x - 3)}$

Step 2.3

Cancel the common factor of $x - 3$ .

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Step 2.3.1

Factor $x - 3$ out of $x^{2} (x - 3)$ .

$\frac{(x - 3) (x + 6)}{x - 1} \cdot \frac{2 x (x - 1)}{(x - 3) x^{2}}$

Step 2.3.2

Cancel the common factor.

$\frac{(x - 3) (x + 6)}{x - 1} \cdot \frac{2 x (x - 1)}{(x - 3) x^{2}}$

Step 2.3.3

Rewrite the expression.

$\frac{x + 6}{x - 1} \cdot \frac{2 x (x - 1)}{x^{2}}$

Step 2.4

Cancel the common factor of $x - 1$ .

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Step 2.4.1

Factor $x - 1$ out of $2 x (x - 1)$ .

$\frac{x + 6}{x - 1} \cdot \frac{(x - 1) (2 x)}{x^{2}}$

Step 2.4.2

Cancel the common factor.

$\frac{x + 6}{x - 1} \cdot \frac{(x - 1) (2 x)}{x^{2}}$

Step 2.4.3

Rewrite the expression.

$(x + 6) \cdot \frac{2 x}{x^{2}}$

Step 2.5

Cancel the common factor of $x$ and $x^{2}$ .

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Step 2.5.1

Factor $x$ out of $2 x$ .

$(x + 6) \cdot \frac{x \cdot 2}{x^{2}}$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $x$ out of $x^{2}$ .

$(x + 6) \cdot \frac{x \cdot 2}{x \cdot x}$

Step 2.5.2.2

Cancel the common factor.

$(x + 6) \cdot \frac{x \cdot 2}{x \cdot x}$

Step 2.5.2.3

Rewrite the expression.

$(x + 6) \cdot \frac{2}{x}$

Step 2.6

Multiply $x + 6$ by $\frac{2}{x}$ .

$\frac{(x + 6) \cdot 2}{x}$

Step 2.7

Move $2$ to the left of $x + 6$ .

$\frac{2 (x + 6)}{x}$