Algebra Examples

$\frac{4 x^{2} - 324}{x^{2} + 4 x - 45} \cdot \frac{4}{- x^{2} - 5 x} \div \frac{9 - x}{25 - x^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 x^{2} - 324}{x^{2} + 4 x - 45} \cdot \frac{4}{- x^{2} - 5 x} \frac{25 - x^{2}}{9 - x}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $4$ out of $4 x^{2} - 324$ .

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Step 2.1.1

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

$\frac{4 (x^{2}) - 324}{x^{2} + 4 x - 45} \cdot \frac{4}{- x^{2} - 5 x} \frac{25 - x^{2}}{9 - x}$

Step 2.1.2

Factor $4$ out of $- 324$ .

$\frac{4 x^{2} + 4 \cdot - 81}{x^{2} + 4 x - 45} \cdot \frac{4}{- x^{2} - 5 x} \frac{25 - x^{2}}{9 - x}$

Step 2.1.3

Factor $4$ out of $4 x^{2} + 4 \cdot - 81$ .

$\frac{4 (x^{2} - 81)}{x^{2} + 4 x - 45} \cdot \frac{4}{- x^{2} - 5 x} \frac{25 - x^{2}}{9 - x}$

Step 2.2

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

$\frac{4 (x^{2} - 9^{2})}{x^{2} + 4 x - 45} \cdot \frac{4}{- x^{2} - 5 x} \frac{25 - x^{2}}{9 - x}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 9$ .

$\frac{4 (x + 9) (x - 9)}{x^{2} + 4 x - 45} \cdot \frac{4}{- x^{2} - 5 x} \frac{25 - x^{2}}{9 - x}$

Step 3

Factor $x^{2} + 4 x - 45$ 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 $- 45$ and whose sum is $4$ .

$- 5, 9$

Step 3.2

Write the factored form using these integers.

$\frac{4 (x + 9) (x - 9)}{(x - 5) (x + 9)} \cdot \frac{4}{- x^{2} - 5 x} \frac{25 - x^{2}}{9 - x}$

Step 4

Simplify terms.

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Step 4.1

Factor $x$ out of $- x^{2} - 5 x$ .

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Step 4.1.1

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

$\frac{4 (x + 9) (x - 9)}{(x - 5) (x + 9)} \cdot \frac{4}{x (- x) - 5 x} \frac{25 - x^{2}}{9 - x}$

Step 4.1.2

Factor $x$ out of $- 5 x$ .

$\frac{4 (x + 9) (x - 9)}{(x - 5) (x + 9)} \cdot \frac{4}{x (- x) + x \cdot - 5} \frac{25 - x^{2}}{9 - x}$

Step 4.1.3

Factor $x$ out of $x (- x) + x \cdot - 5$ .

$\frac{4 (x + 9) (x - 9)}{(x - 5) (x + 9)} \cdot \frac{4}{x (- x - 5)} \frac{25 - x^{2}}{9 - x}$

Step 4.2

Cancel the common factor of $x + 9$ .

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Step 4.2.1

Cancel the common factor.

$\frac{4 (x + 9) (x - 9)}{(x - 5) (x + 9)} \cdot \frac{4}{x (- x - 5)} \frac{25 - x^{2}}{9 - x}$

Step 4.2.2

Rewrite the expression.

$\frac{4 (x - 9)}{x - 5} \cdot \frac{4}{x (- x - 5)} \frac{25 - x^{2}}{9 - x}$

Step 5

Multiply $\frac{4 (x - 9)}{x - 5} \cdot \frac{4}{x (- x - 5)}$ .

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Step 5.1

Multiply $\frac{4 (x - 9)}{x - 5}$ by $\frac{4}{x (- x - 5)}$ .

$\frac{4 (x - 9) \cdot 4}{(x - 5) (x (- x - 5))} \cdot \frac{25 - x^{2}}{9 - x}$

Step 5.2

Multiply $4$ by $4$ .

$\frac{16 (x - 9)}{(x - 5) (x (- x - 5))} \cdot \frac{25 - x^{2}}{9 - x}$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{16 (x - 9)}{(x - 5) x (- x - 5)} \cdot \frac{5^{2} - x^{2}}{9 - x}$

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

$\frac{16 (x - 9)}{(x - 5) x (- x - 5)} \cdot \frac{(5 + x) (5 - x)}{9 - x}$

Step 7

Simplify terms.

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Step 7.1

Multiply $\frac{16 (x - 9)}{(x - 5) x (- x - 5)}$ by $\frac{(5 + x) (5 - x)}{9 - x}$ .

$\frac{16 (x - 9) ((5 + x) (5 - x))}{(x - 5) x (- x - 5) (9 - x)}$

Step 7.2

Cancel the common factor of $x - 9$ and $9 - x$ .

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Step 7.2.1

Factor $- 1$ out of $x$ .

$\frac{16 (- 1 (- x) - 9) ((5 + x) (5 - x))}{(x - 5) x (- x - 5) (9 - x)}$

Step 7.2.2

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{16 (- 1 (- x) - 1 (9)) ((5 + x) (5 - x))}{(x - 5) x (- x - 5) (9 - x)}$

Step 7.2.3

Factor $- 1$ out of $- 1 (- x) - 1 (9)$ .

$\frac{16 (- 1 (- x + 9)) ((5 + x) (5 - x))}{(x - 5) x (- x - 5) (9 - x)}$

Step 7.2.4

Reorder terms.

$\frac{16 (- 1 (- x + 9)) ((5 + x) (5 - x))}{(x - 5) x (- x - 5) (- x + 9)}$

Step 7.2.5

Cancel the common factor.

$\frac{16 (- 1 (- x + 9)) ((5 + x) (5 - x))}{(x - 5) x (- x - 5) (- x + 9)}$

Step 7.2.6

Rewrite the expression.

$\frac{16 \cdot (- 1) ((5 + x) (5 - x))}{(x - 5) x (- x - 5)}$

Step 7.3

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

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Step 7.3.1

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{16 \cdot (- 1) ((- 1 (- 5) + x) (5 - x))}{(x - 5) x (- x - 5)}$

Step 7.3.2

Factor $- 1$ out of $x$ .

$\frac{16 \cdot (- 1) ((- 1 (- 5) - 1 (- x)) (5 - x))}{(x - 5) x (- x - 5)}$

Step 7.3.3

Factor $- 1$ out of $- 1 (- 5) - 1 (- x)$ .

$\frac{16 \cdot (- 1) (- 1 (- 5 - x) (5 - x))}{(x - 5) x (- x - 5)}$

Step 7.3.4

Reorder terms.

$\frac{16 \cdot (- 1) (- 1 (- x - 5) (5 - x))}{(x - 5) x (- x - 5)}$

Step 7.3.5

Cancel the common factor.

$\frac{16 \cdot (- 1) (- 1 (- x - 5) (5 - x))}{(x - 5) x (- x - 5)}$

Step 7.3.6

Rewrite the expression.

$\frac{16 \cdot (- 1) (- 1 (5 - x))}{(x - 5) x}$

Step 7.4

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

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Step 7.4.1

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{16 \cdot (- 1) (- 1 (- 1 (- 5) - x))}{(x - 5) x}$

Step 7.4.2

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

$\frac{16 \cdot (- 1) (- 1 (- 1 (- 5) - (x)))}{(x - 5) x}$

Step 7.4.3

Factor $- 1$ out of $- 1 (- 5) - (x)$ .

$\frac{16 \cdot (- 1) (- 1 (- 1 (- 5 + x)))}{(x - 5) x}$

Step 7.4.4

Reorder terms.

$\frac{16 \cdot (- 1) (- 1 (- 1 (x - 5)))}{(x - 5) x}$

Step 7.4.5

Cancel the common factor.

$\frac{16 \cdot (- 1) (- 1 (- 1 (x - 5)))}{(x - 5) x}$

Step 7.4.6

Rewrite the expression.

$\frac{16 \cdot (- 1) (- 1 \cdot (- 1))}{x}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $16$ by $- 1$ .

$\frac{- 16 (- 1 \cdot (- 1))}{x}$

Step 8.2

Multiply $- 1$ by $- 16$ .

$\frac{16 \cdot - 1}{x}$

Step 8.3

Multiply $16$ by $- 1$ .

$\frac{- 16}{x}$

Step 9

Move the negative in front of the fraction.

$- \frac{16}{x}$