Algebra Examples

$\frac{a^{2} + 3 a - 18}{a^{2} - 5 a + 6} \cdot \frac{5 a - 10}{a^{2} - 36}$

Step 1

Factor $a^{2} + 3 a - 18$ 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.

$\frac{(a - 3) (a + 6)}{a^{2} - 5 a + 6} \cdot \frac{5 a - 10}{a^{2} - 36}$

Step 2

Factor $a^{2} - 5 a + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 2.2

Write the factored form using these integers.

$\frac{(a - 3) (a + 6)}{(a - 3) (a - 2)} \cdot \frac{5 a - 10}{a^{2} - 36}$

Step 3

Factor $5$ out of $5 a - 10$ .

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Step 3.1

Factor $5$ out of $5 a$ .

$\frac{(a - 3) (a + 6)}{(a - 3) (a - 2)} \cdot \frac{5 (a) - 10}{a^{2} - 36}$

Step 3.2

Factor $5$ out of $- 10$ .

$\frac{(a - 3) (a + 6)}{(a - 3) (a - 2)} \cdot \frac{5 a + 5 \cdot - 2}{a^{2} - 36}$

Step 3.3

Factor $5$ out of $5 a + 5 \cdot - 2$ .

$\frac{(a - 3) (a + 6)}{(a - 3) (a - 2)} \cdot \frac{5 (a - 2)}{a^{2} - 36}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $36$ as $6^{2}$ .

$\frac{(a - 3) (a + 6)}{(a - 3) (a - 2)} \cdot \frac{5 (a - 2)}{a^{2} - 6^{2}}$

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 $b = 6$ .

$\frac{(a - 3) (a + 6)}{(a - 3) (a - 2)} \cdot \frac{5 (a - 2)}{(a + 6) (a - 6)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $a + 6$ .

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Step 5.1.1

Factor $a + 6$ out of $(a - 3) (a + 6)$ .

$\frac{(a + 6) (a - 3)}{(a - 3) (a - 2)} \cdot \frac{5 (a - 2)}{(a + 6) (a - 6)}$

Step 5.1.2

Cancel the common factor.

$\frac{(a + 6) (a - 3)}{(a - 3) (a - 2)} \cdot \frac{5 (a - 2)}{(a + 6) (a - 6)}$

Step 5.1.3

Rewrite the expression.

$\frac{a - 3}{(a - 3) (a - 2)} \cdot \frac{5 (a - 2)}{a - 6}$

Step 5.2

Cancel the common factor of $a - 2$ .

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Step 5.2.1

Factor $a - 2$ out of $(a - 3) (a - 2)$ .

$\frac{a - 3}{(a - 2) (a - 3)} \cdot \frac{5 (a - 2)}{a - 6}$

Step 5.2.2

Factor $a - 2$ out of $5 (a - 2)$ .

$\frac{a - 3}{(a - 2) (a - 3)} \cdot \frac{(a - 2) \cdot 5}{a - 6}$

Step 5.2.3

Cancel the common factor.

$\frac{a - 3}{(a - 2) (a - 3)} \cdot \frac{(a - 2) \cdot 5}{a - 6}$

Step 5.2.4

Rewrite the expression.

$\frac{a - 3}{a - 3} \cdot \frac{5}{a - 6}$

Step 5.3

Multiply $\frac{a - 3}{a - 3}$ by $\frac{5}{a - 6}$ .

$\frac{(a - 3) \cdot 5}{(a - 3) (a - 6)}$

Step 5.4

Cancel the common factor of $a - 3$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(a - 3) \cdot 5}{(a - 3) (a - 6)}$

Step 5.4.2

Rewrite the expression.

$\frac{5}{a - 6}$