Basic Math Examples

$\frac{c^{2} - 14 c + 45}{c^{2} - 3 c - 10} \cdot \frac{c^{2} - 4}{c^{2} - 81}$

Step 1

Factor $c^{2} - 14 c + 45$ 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 $45$ and whose sum is $- 14$ .

$- 9, - 5$

Step 1.2

Write the factored form using these integers.

$\frac{(c - 9) (c - 5)}{c^{2} - 3 c - 10} \cdot \frac{c^{2} - 4}{c^{2} - 81}$

Step 2

Factor $c^{2} - 3 c - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(c - 9) (c - 5)}{(c - 5) (c + 2)} \cdot \frac{c^{2} - 4}{c^{2} - 81}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{(c - 9) (c - 5)}{(c - 5) (c + 2)} \cdot \frac{c^{2} - 2^{2}}{c^{2} - 81}$

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 $a = c$ and $b = 2$ .

$\frac{(c - 9) (c - 5)}{(c - 5) (c + 2)} \cdot \frac{(c + 2) (c - 2)}{c^{2} - 81}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $81$ as $9^{2}$ .

$\frac{(c - 9) (c - 5)}{(c - 5) (c + 2)} \cdot \frac{(c + 2) (c - 2)}{c^{2} - 9^{2}}$

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 $a = c$ and $b = 9$ .

$\frac{(c - 9) (c - 5)}{(c - 5) (c + 2)} \cdot \frac{(c + 2) (c - 2)}{(c + 9) (c - 9)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $c - 9$ .

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Step 5.1.1

Factor $c - 9$ out of $(c + 9) (c - 9)$ .

$\frac{(c - 9) (c - 5)}{(c - 5) (c + 2)} \cdot \frac{(c + 2) (c - 2)}{(c - 9) (c + 9)}$

Step 5.1.2

Cancel the common factor.

$\frac{(c - 9) (c - 5)}{(c - 5) (c + 2)} \cdot \frac{(c + 2) (c - 2)}{(c - 9) (c + 9)}$

Step 5.1.3

Rewrite the expression.

$\frac{c - 5}{(c - 5) (c + 2)} \cdot \frac{(c + 2) (c - 2)}{c + 9}$

Step 5.2

Cancel the common factor of $c + 2$ .

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Step 5.2.1

Factor $c + 2$ out of $(c - 5) (c + 2)$ .

$\frac{c - 5}{(c + 2) (c - 5)} \cdot \frac{(c + 2) (c - 2)}{c + 9}$

Step 5.2.2

Cancel the common factor.

$\frac{c - 5}{(c + 2) (c - 5)} \cdot \frac{(c + 2) (c - 2)}{c + 9}$

Step 5.2.3

Rewrite the expression.

$\frac{c - 5}{c - 5} \cdot \frac{c - 2}{c + 9}$

Step 5.3

Multiply $\frac{c - 5}{c - 5}$ by $\frac{c - 2}{c + 9}$ .

$\frac{(c - 5) (c - 2)}{(c - 5) (c + 9)}$

Step 5.4

Cancel the common factor of $c - 5$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(c - 5) (c - 2)}{(c - 5) (c + 9)}$

Step 5.4.2

Rewrite the expression.

$\frac{c - 2}{c + 9}$