Basic Math Examples

$\frac{36 - n^{2}}{n^{2} - 3 n - 54} \cdot \frac{n^{2} - 7 n - 30}{n^{2} - 3 n - 18}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{6^{2} - n^{2}}{n^{2} - 3 n - 54} \cdot \frac{n^{2} - 7 n - 30}{n^{2} - 3 n - 18}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = n$ .

$\frac{(6 + n) (6 - n)}{n^{2} - 3 n - 54} \cdot \frac{n^{2} - 7 n - 30}{n^{2} - 3 n - 18}$

Step 2

Factor $n^{2} - 3 n - 54$ 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 $- 54$ and whose sum is $- 3$ .

$- 9, 6$

Step 2.2

Write the factored form using these integers.

$\frac{(6 + n) (6 - n)}{(n - 9) (n + 6)} \cdot \frac{n^{2} - 7 n - 30}{n^{2} - 3 n - 18}$

Step 3

Factor $n^{2} - 7 n - 30$ 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 $- 30$ and whose sum is $- 7$ .

$- 10, 3$

Step 3.2

Write the factored form using these integers.

$\frac{(6 + n) (6 - n)}{(n - 9) (n + 6)} \cdot \frac{(n - 10) (n + 3)}{n^{2} - 3 n - 18}$

Step 4

Factor $n^{2} - 3 n - 18$ using the AC method.

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Step 4.1

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 $- 3$ .

$- 6, 3$

Step 4.2

Write the factored form using these integers.

$\frac{(6 + n) (6 - n)}{(n - 9) (n + 6)} \cdot \frac{(n - 10) (n + 3)}{(n - 6) (n + 3)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $6 + n$ and $n + 6$ .

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Step 5.1.1

Reorder terms.

$\frac{(n + 6) (6 - n)}{(n - 9) (n + 6)} \cdot \frac{(n - 10) (n + 3)}{(n - 6) (n + 3)}$

Step 5.1.2

Cancel the common factor.

$\frac{(n + 6) (6 - n)}{(n - 9) (n + 6)} \cdot \frac{(n - 10) (n + 3)}{(n - 6) (n + 3)}$

Step 5.1.3

Rewrite the expression.

$\frac{6 - n}{n - 9} \cdot \frac{(n - 10) (n + 3)}{(n - 6) (n + 3)}$

Step 5.2

Cancel the common factor of $n + 3$ .

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Step 5.2.1

Cancel the common factor.

$\frac{6 - n}{n - 9} \cdot \frac{(n - 10) (n + 3)}{(n - 6) (n + 3)}$

Step 5.2.2

Rewrite the expression.

$\frac{6 - n}{n - 9} \cdot \frac{n - 10}{n - 6}$

Step 5.3

Multiply $\frac{6 - n}{n - 9}$ by $\frac{n - 10}{n - 6}$ .

$\frac{(6 - n) (n - 10)}{(n - 9) (n - 6)}$

Step 5.4

Cancel the common factor of $6 - n$ and $n - 6$ .

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Step 5.4.1

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{(- 1 (- 6) - n) (n - 10)}{(n - 9) (n - 6)}$

Step 5.4.2

Factor $- 1$ out of $- n$ .

$\frac{(- 1 (- 6) - (n)) (n - 10)}{(n - 9) (n - 6)}$

Step 5.4.3

Factor $- 1$ out of $- 1 (- 6) - (n)$ .

$\frac{- 1 (- 6 + n) (n - 10)}{(n - 9) (n - 6)}$

Step 5.4.4

Reorder terms.

$\frac{- 1 (n - 6) (n - 10)}{(n - 9) (n - 6)}$

Step 5.4.5

Cancel the common factor.

$\frac{- 1 (n - 6) (n - 10)}{(n - 9) (n - 6)}$

Step 5.4.6

Rewrite the expression.

$\frac{- 1 (n - 10)}{n - 9}$

Step 5.5

Move the negative in front of the fraction.

$- \frac{n - 10}{n - 9}$