Basic Math Examples

$\frac{a^{2} - 6 a + 8}{a^{2} - a - 12} \cdot \frac{a^{2} - 9}{a^{2} - 4}$

Step 1

Factor $a^{2} - 6 a + 8$ 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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 1.2

Write the factored form using these integers.

$\frac{(a - 4) (a - 2)}{a^{2} - a - 12} \cdot \frac{a^{2} - 9}{a^{2} - 4}$

Step 2

Factor $a^{2} - a - 12$ using the AC method.

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Step 2.1

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 2.2

Write the factored form using these integers.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.2

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

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $a - 2$ .

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Step 5.1.1

Factor $a - 2$ out of $(a - 4) (a - 2)$ .

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

Step 5.1.2

Factor $a - 2$ out of $(a + 2) (a - 2)$ .

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

Step 5.1.3

Cancel the common factor.

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

Step 5.1.4

Rewrite the expression.

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

Step 5.2

Cancel the common factor of $a + 3$ .

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Step 5.2.1

Factor $a + 3$ out of $(a - 4) (a + 3)$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$\frac{a - 4}{a - 4} \cdot \frac{a - 3}{a + 2}$

Step 5.3

Multiply $\frac{a - 4}{a - 4}$ by $\frac{a - 3}{a + 2}$ .

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

Step 5.4

Cancel the common factor of $a - 4$ .

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Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Rewrite the expression.

$\frac{a - 3}{a + 2}$