Finite Math Examples

$\frac{5 a^{2} + 20 a}{a^{3} - 2 a^{2}} \cdot \frac{x (a^{2} - a - 12)}{a - 16}$

Step 1

Factor $5 a$ out of $5 a^{2} + 20 a$ .

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Step 1.1

Factor $5 a$ out of $5 a^{2}$ .

$\frac{5 a (a) + 20 a}{a^{3} - 2 a^{2}} \cdot \frac{x (a^{2} - a - 12)}{a - 16}$

Step 1.2

Factor $5 a$ out of $20 a$ .

$\frac{5 a (a) + 5 a (4)}{a^{3} - 2 a^{2}} \cdot \frac{x (a^{2} - a - 12)}{a - 16}$

Step 1.3

Factor $5 a$ out of $5 a (a) + 5 a (4)$ .

$\frac{5 a (a + 4)}{a^{3} - 2 a^{2}} \cdot \frac{x (a^{2} - a - 12)}{a - 16}$

Step 2

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

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Step 2.1

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

$\frac{5 a (a + 4)}{a^{2} a - 2 a^{2}} \cdot \frac{x (a^{2} - a - 12)}{a - 16}$

Step 2.2

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

$\frac{5 a (a + 4)}{a^{2} a + a^{2} \cdot - 2} \cdot \frac{x (a^{2} - a - 12)}{a - 16}$

Step 2.3

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

$\frac{5 a (a + 4)}{a^{2} (a - 2)} \cdot \frac{x (a^{2} - a - 12)}{a - 16}$

Step 3

Factor $a^{2} - a - 12$ 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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 3.2

Write the factored form using these integers.

$\frac{5 a (a + 4)}{a^{2} (a - 2)} \cdot \frac{x ((a - 4) (a + 3))}{a - 16}$

$\frac{5 a (a + 4)}{a^{2} (a - 2)} \cdot \frac{x (a - 4) (a + 3)}{a - 16}$

Step 4

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

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Step 4.1

Factor $a$ out of $5 a (a + 4)$ .

$\frac{a (5 (a + 4))}{a^{2} (a - 2)} \cdot \frac{x (a - 4) (a + 3)}{a - 16}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

$\frac{a (5 (a + 4))}{a (a (a - 2))} \cdot \frac{x (a - 4) (a + 3)}{a - 16}$

Step 4.2.2

Cancel the common factor.

$\frac{a (5 (a + 4))}{a (a (a - 2))} \cdot \frac{x (a - 4) (a + 3)}{a - 16}$

Step 4.2.3

Rewrite the expression.

$\frac{5 (a + 4)}{a (a - 2)} \cdot \frac{x (a - 4) (a + 3)}{a - 16}$

Step 5

Multiply $\frac{5 (a + 4)}{a (a - 2)}$ by $\frac{x (a - 4) (a + 3)}{a - 16}$ .

$\frac{5 (a + 4) (x (a - 4) (a + 3))}{a (a - 2) (a - 16)}$

Step 6

Reorder factors in $\frac{5 (a + 4) x (a - 4) (a + 3)}{a (a - 2) (a - 16)}$ .

$\frac{5 x (a + 4) (a - 4) (a + 3)}{a (a - 2) (a - 16)}$