Algebra Examples

$\frac{x^{2} + x - 20}{4 x^{2} - 40 x + 96} \cdot \frac{3}{25 - x^{2}} \div \frac{x + 6}{36 - x^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + x - 20}{4 x^{2} - 40 x + 96} \cdot \frac{3}{25 - x^{2}} \frac{36 - x^{2}}{x + 6}$

Step 2

Factor $x^{2} + x - 20$ 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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 5)}{4 x^{2} - 40 x + 96} \cdot \frac{3}{25 - x^{2}} \frac{36 - x^{2}}{x + 6}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $4$ out of $4 x^{2} - 40 x + 96$ .

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Step 3.1.1

Factor $4$ out of $4 x^{2}$ .

$\frac{(x - 4) (x + 5)}{4 (x^{2}) - 40 x + 96} \cdot \frac{3}{25 - x^{2}} \frac{36 - x^{2}}{x + 6}$

Step 3.1.2

Factor $4$ out of $- 40 x$ .

$\frac{(x - 4) (x + 5)}{4 (x^{2}) + 4 (- 10 x) + 96} \cdot \frac{3}{25 - x^{2}} \frac{36 - x^{2}}{x + 6}$

Step 3.1.3

Factor $4$ out of $96$ .

$\frac{(x - 4) (x + 5)}{4 x^{2} + 4 (- 10 x) + 4 \cdot 24} \cdot \frac{3}{25 - x^{2}} \frac{36 - x^{2}}{x + 6}$

Step 3.1.4

Factor $4$ out of $4 x^{2} + 4 (- 10 x)$ .

$\frac{(x - 4) (x + 5)}{4 (x^{2} - 10 x) + 4 \cdot 24} \cdot \frac{3}{25 - x^{2}} \frac{36 - x^{2}}{x + 6}$

Step 3.1.5

Factor $4$ out of $4 (x^{2} - 10 x) + 4 \cdot 24$ .

$\frac{(x - 4) (x + 5)}{4 (x^{2} - 10 x + 24)} \cdot \frac{3}{25 - x^{2}} \frac{36 - x^{2}}{x + 6}$

Step 3.2

Factor $x^{2} - 10 x + 24$ using the AC method.

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Step 3.2.1

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 $24$ and whose sum is $- 10$ .

$- 6, - 4$

Step 3.2.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 5)}{4 ((x - 6) (x - 4))} \cdot \frac{3}{25 - x^{2}} \frac{36 - x^{2}}{x + 6}$

$\frac{(x - 4) (x + 5)}{4 (x - 6) (x - 4)} \cdot \frac{3}{25 - x^{2}} \frac{36 - x^{2}}{x + 6}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $25$ as $5^{2}$ .

$\frac{(x - 4) (x + 5)}{4 (x - 6) (x - 4)} \cdot \frac{3}{5^{2} - x^{2}} \frac{36 - x^{2}}{x + 6}$

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 $a = 5$ and $b = x$ .

$\frac{(x - 4) (x + 5)}{4 (x - 6) (x - 4)} \cdot \frac{3}{(5 + x) (5 - x)} \frac{36 - x^{2}}{x + 6}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x - 4$ .

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Step 5.1.1

Cancel the common factor.

$\frac{(x - 4) (x + 5)}{4 (x - 6) (x - 4)} \cdot \frac{3}{(5 + x) (5 - x)} \frac{36 - x^{2}}{x + 6}$

Step 5.1.2

Rewrite the expression.

$\frac{x + 5}{4 (x - 6)} \cdot \frac{3}{(5 + x) (5 - x)} \frac{36 - x^{2}}{x + 6}$

Step 5.2

Multiply $\frac{x + 5}{4 (x - 6)}$ by $\frac{3}{(5 + x) (5 - x)}$ .

$\frac{(x + 5) \cdot 3}{4 (x - 6) ((5 + x) (5 - x))} \cdot \frac{36 - x^{2}}{x + 6}$

Step 5.3

Cancel the common factor of $x + 5$ and $5 + x$ .

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Step 5.3.1

Reorder terms.

$\frac{(x + 5) \cdot 3}{4 (x - 6) ((x + 5) (5 - x))} \cdot \frac{36 - x^{2}}{x + 6}$

Step 5.3.2

Cancel the common factor.

$\frac{(x + 5) \cdot 3}{4 (x - 6) ((x + 5) (5 - x))} \cdot \frac{36 - x^{2}}{x + 6}$

Step 5.3.3

Rewrite the expression.

$\frac{3}{4 (x - 6) (5 - x)} \cdot \frac{36 - x^{2}}{x + 6}$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{3}{4 (x - 6) (5 - x)} \cdot \frac{6^{2} - x^{2}}{x + 6}$

Step 6.2

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 $b = x$ .

$\frac{3}{4 (x - 6) (5 - x)} \cdot \frac{(6 + x) (6 - x)}{x + 6}$

Step 7

Simplify terms.

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Step 7.1

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

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Step 7.1.1

Reorder terms.

$\frac{3}{4 (x - 6) (5 - x)} \cdot \frac{(x + 6) (6 - x)}{x + 6}$

Step 7.1.2

Cancel the common factor.

$\frac{3}{4 (x - 6) (5 - x)} \cdot \frac{(x + 6) (6 - x)}{x + 6}$

Step 7.1.3

Divide $6 - x$ by $1$ .

$\frac{3}{4 (x - 6) (5 - x)} (6 - x)$

Step 7.2

Multiply $\frac{3}{4 (x - 6) (5 - x)}$ by $6 - x$ .

$\frac{3 (6 - x)}{4 (x - 6) (5 - x)}$

Step 7.3

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

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Step 7.3.1

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

$\frac{3 (- 1 (- 6) - x)}{4 (x - 6) (5 - x)}$

Step 7.3.2

Factor $- 1$ out of $- x$ .

$\frac{3 (- 1 (- 6) - (x))}{4 (x - 6) (5 - x)}$

Step 7.3.3

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

$\frac{3 (- 1 (- 6 + x))}{4 (x - 6) (5 - x)}$

Step 7.3.4

Reorder terms.

$\frac{3 (- 1 (x - 6))}{4 (x - 6) (5 - x)}$

Step 7.3.5

Cancel the common factor.

$\frac{3 (- 1 (x - 6))}{4 (x - 6) (5 - x)}$

Step 7.3.6

Rewrite the expression.

$\frac{3 \cdot (- 1)}{4 (5 - x)}$

Step 8

Multiply $3$ by $- 1$ .

$\frac{- 3}{4 (5 - x)}$

Step 9

Move the negative in front of the fraction.

$- \frac{3}{4 (5 - x)}$