Algebra Examples

$\frac{45 x^{2}}{x - 9} \cdot \frac{x^{2} - 5 x - 36}{3 x^{3} + 12 x^{2}}$

Step 1

Factor $x^{2} - 5 x - 36$ 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 $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Step 1.2

Write the factored form using these integers.

$\frac{45 x^{2}}{x - 9} \cdot \frac{(x - 9) (x + 4)}{3 x^{3} + 12 x^{2}}$

Step 2

Simplify terms.

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Step 2.1

Factor $3 x^{2}$ out of $3 x^{3} + 12 x^{2}$ .

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Step 2.1.1

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

$\frac{45 x^{2}}{x - 9} \cdot \frac{(x - 9) (x + 4)}{3 x^{2} (x) + 12 x^{2}}$

Step 2.1.2

Factor $3 x^{2}$ out of $12 x^{2}$ .

$\frac{45 x^{2}}{x - 9} \cdot \frac{(x - 9) (x + 4)}{3 x^{2} (x) + 3 x^{2} (4)}$

Step 2.1.3

Factor $3 x^{2}$ out of $3 x^{2} (x) + 3 x^{2} (4)$ .

$\frac{45 x^{2}}{x - 9} \cdot \frac{(x - 9) (x + 4)}{3 x^{2} (x + 4)}$

Step 2.2

Cancel the common factor of $3 x^{2}$ .

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Step 2.2.1

Factor $3 x^{2}$ out of $45 x^{2}$ .

$\frac{3 x^{2} (15)}{x - 9} \cdot \frac{(x - 9) (x + 4)}{3 x^{2} (x + 4)}$

Step 2.2.2

Cancel the common factor.

$\frac{3 x^{2} \cdot 15}{x - 9} \cdot \frac{(x - 9) (x + 4)}{3 x^{2} (x + 4)}$

Step 2.2.3

Rewrite the expression.

$\frac{15}{x - 9} \cdot \frac{(x - 9) (x + 4)}{x + 4}$

Step 2.3

Cancel the common factor of $x - 9$ .

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Step 2.3.1

Cancel the common factor.

$\frac{15}{x - 9} \cdot \frac{(x - 9) (x + 4)}{x + 4}$

Step 2.3.2

Rewrite the expression.

$15 \cdot \frac{x + 4}{x + 4}$

Step 2.4

Combine $15$ and $\frac{x + 4}{x + 4}$ .

$\frac{15 (x + 4)}{x + 4}$

Step 2.5

Cancel the common factor of $x + 4$ .

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Step 2.5.1

Cancel the common factor.

$\frac{15 (x + 4)}{x + 4}$

Step 2.5.2

Divide $15$ by $1$ .

$15$