Finite Math Examples

$\frac{x^{2} + 12 x + 20}{4 x^{2} - 9} \cdot \frac{6 x^{3} - 9 x^{2}}{x^{3} + 10 x^{2}} \cdot (2 x + 3)$

Step 1

Factor $x^{2} + 12 x + 20$ 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 $20$ and whose sum is $12$ .

$2, 10$

Step 1.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 10)}{4 x^{2} - 9} \cdot \frac{6 x^{3} - 9 x^{2}}{x^{3} + 10 x^{2}} \cdot (2 x + 3)$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{(x + 2) (x + 10)}{{(2 x)}^{2} - 9} \cdot \frac{6 x^{3} - 9 x^{2}}{x^{3} + 10 x^{2}} \cdot (2 x + 3)$

Step 2.2

Rewrite $9$ as $3^{2}$ .

$\frac{(x + 2) (x + 10)}{{(2 x)}^{2} - 3^{2}} \cdot \frac{6 x^{3} - 9 x^{2}}{x^{3} + 10 x^{2}} \cdot (2 x + 3)$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 3$ .

$\frac{(x + 2) (x + 10)}{(2 x + 3) (2 x - 3)} \cdot \frac{6 x^{3} - 9 x^{2}}{x^{3} + 10 x^{2}} \cdot (2 x + 3)$

Step 3

Simplify terms.

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Step 3.1

Factor $3 x^{2}$ out of $6 x^{3} - 9 x^{2}$ .

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Step 3.1.1

Factor $3 x^{2}$ out of $6 x^{3}$ .

$\frac{(x + 2) (x + 10)}{(2 x + 3) (2 x - 3)} \cdot \frac{3 x^{2} (2 x) - 9 x^{2}}{x^{3} + 10 x^{2}} \cdot (2 x + 3)$

Step 3.1.2

Factor $3 x^{2}$ out of $- 9 x^{2}$ .

$\frac{(x + 2) (x + 10)}{(2 x + 3) (2 x - 3)} \cdot \frac{3 x^{2} (2 x) + 3 x^{2} (- 3)}{x^{3} + 10 x^{2}} \cdot (2 x + 3)$

Step 3.1.3

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

$\frac{(x + 2) (x + 10)}{(2 x + 3) (2 x - 3)} \cdot \frac{3 x^{2} (2 x - 3)}{x^{3} + 10 x^{2}} \cdot (2 x + 3)$

Step 3.2

Factor $x^{2}$ out of $x^{3} + 10 x^{2}$ .

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Step 3.2.1

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

$\frac{(x + 2) (x + 10)}{(2 x + 3) (2 x - 3)} \cdot \frac{3 x^{2} (2 x - 3)}{x^{2} x + 10 x^{2}} \cdot (2 x + 3)$

Step 3.2.2

Factor $x^{2}$ out of $10 x^{2}$ .

$\frac{(x + 2) (x + 10)}{(2 x + 3) (2 x - 3)} \cdot \frac{3 x^{2} (2 x - 3)}{x^{2} x + x^{2} \cdot 10} \cdot (2 x + 3)$

Step 3.2.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 10$ .

$\frac{(x + 2) (x + 10)}{(2 x + 3) (2 x - 3)} \cdot \frac{3 x^{2} (2 x - 3)}{x^{2} (x + 10)} \cdot (2 x + 3)$

Step 3.3

Combine.

$\frac{(x + 2) (x + 10) (3 x^{2} (2 x - 3))}{(2 x + 3) (2 x - 3) (x^{2} (x + 10))} \cdot (2 x + 3)$

Step 3.4

Cancel the common factor of $2 x + 3$ .

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Step 3.4.1

Factor $2 x + 3$ out of $(2 x + 3) (2 x - 3) x^{2} (x + 10)$ .

$\frac{(x + 2) (x + 10) \cdot 3 x^{2} (2 x - 3)}{(2 x + 3) ((2 x - 3) x^{2} (x + 10))} \cdot (2 x + 3)$

Step 3.4.2

Cancel the common factor.

$\frac{(x + 2) (x + 10) \cdot 3 x^{2} (2 x - 3)}{(2 x + 3) ((2 x - 3) x^{2} (x + 10))} \cdot (2 x + 3)$

Step 3.4.3

Rewrite the expression.

$\frac{(x + 2) (x + 10) \cdot 3 x^{2} (2 x - 3)}{(2 x - 3) x^{2} (x + 10)}$

Step 3.5

Cancel the common factor of $x + 10$ .

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Step 3.5.1

Cancel the common factor.

$\frac{(x + 2) (x + 10) \cdot 3 x^{2} (2 x - 3)}{(2 x - 3) x^{2} (x + 10)}$

Step 3.5.2

Rewrite the expression.

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

Step 3.6

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

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Step 3.6.1

Cancel the common factor.

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

Step 3.6.2

Rewrite the expression.

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

Step 3.7

Cancel the common factor of $2 x - 3$ .

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Step 3.7.1

Cancel the common factor.

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

Step 3.7.2

Divide $(x + 2) \cdot 3$ by $1$ .

$(x + 2) \cdot 3$

Step 3.8

Apply the distributive property.

$x \cdot 3 + 2 \cdot 3$

Step 3.9

Simplify the expression.

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Step 3.9.1

Move $3$ to the left of $x$ .

$3 \cdot x + 2 \cdot 3$

Step 3.9.2

Multiply $2$ by $3$ .

$3 x + 6$