Algebra Examples

$\frac{9 x}{x^{2} - 25} \cdot \frac{x^{2} + 5 x}{2 x - 4} \cdot \frac{x^{2} + 3 x - 10}{3 x^{4}}$

Step 1

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

$- 2, 5$

Step 1.2

Write the factored form using these integers.

$\frac{9 x}{x^{2} - 25} \cdot \frac{x^{2} + 5 x}{2 x - 4} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{9 x}{x^{2} - 5^{2}} \cdot \frac{x^{2} + 5 x}{2 x - 4} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 2.2

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 $b = 5$ .

$\frac{9 x}{(x + 5) (x - 5)} \cdot \frac{x^{2} + 5 x}{2 x - 4} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

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

$\frac{9 x}{(x + 5) (x - 5)} \cdot \frac{x \cdot x + 5 x}{2 x - 4} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 3.1.2

Factor $x$ out of $5 x$ .

$\frac{9 x}{(x + 5) (x - 5)} \cdot \frac{x \cdot x + x \cdot 5}{2 x - 4} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 3.1.3

Factor $x$ out of $x \cdot x + x \cdot 5$ .

$\frac{9 x}{(x + 5) (x - 5)} \cdot \frac{x (x + 5)}{2 x - 4} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 3.2

Factor $2$ out of $2 x - 4$ .

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Step 3.2.1

Factor $2$ out of $2 x$ .

$\frac{9 x}{(x + 5) (x - 5)} \cdot \frac{x (x + 5)}{2 (x) - 4} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 3.2.2

Factor $2$ out of $- 4$ .

$\frac{9 x}{(x + 5) (x - 5)} \cdot \frac{x (x + 5)}{2 x + 2 \cdot - 2} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 3.2.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$\frac{9 x}{(x + 5) (x - 5)} \cdot \frac{x (x + 5)}{2 (x - 2)} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 3.3

Cancel the common factor of $x + 5$ .

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Step 3.3.1

Factor $x + 5$ out of $x (x + 5)$ .

$\frac{9 x}{(x + 5) (x - 5)} \cdot \frac{(x + 5) x}{2 (x - 2)} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 3.3.2

Cancel the common factor.

$\frac{9 x}{(x + 5) (x - 5)} \cdot \frac{(x + 5) x}{2 (x - 2)} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 3.3.3

Rewrite the expression.

$\frac{9 x}{x - 5} \cdot \frac{x}{2 (x - 2)} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 3.4

Multiply $\frac{9 x}{x - 5}$ by $\frac{x}{2 (x - 2)}$ .

$\frac{9 x \cdot x}{(x - 5) (2 (x - 2))} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 4

Raise $x$ to the power of $1$ .

$\frac{9 (x^{1} x)}{(x - 5) (2 (x - 2))} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 5

Raise $x$ to the power of $1$ .

$\frac{9 (x^{1} x^{1})}{(x - 5) (2 (x - 2))} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 x^{1 + 1}}{(x - 5) (2 (x - 2))} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 7

Add $1$ and $1$ .

$\frac{9 x^{2}}{(x - 5) (2 (x - 2))} \cdot \frac{(x - 2) (x + 5)}{3 x^{4}}$

Step 8

Combine.

$\frac{9 x^{2} ((x - 2) (x + 5))}{(x - 5) (2 (x - 2)) (3 x^{4})}$

Step 9

Cancel the common factor of $9$ and $3$ .

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Step 9.1

Factor $3$ out of $9 x^{2} ((x - 2) (x + 5))$ .

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

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $3$ out of $(x - 5) (2 (x - 2)) (3 x^{4})$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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

Step 10

Cancel the common factor of $x^{2}$ and $x^{4}$ .

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Step 10.1

Factor $x^{2}$ out of $3 x^{2} ((x - 2) (x + 5))$ .

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

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $x^{2}$ out of $(x - 5) (2 (x - 2)) (x^{4})$ .

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

Step 10.2.2

Cancel the common factor.

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

Step 10.2.3

Rewrite the expression.

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

Step 11

Cancel the common factor of $x - 2$ .

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Step 11.1

Cancel the common factor.

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

Step 11.2

Rewrite the expression.

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

Step 12

Simplify the expression.

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Step 12.1

Move $2$ to the left of $x - 5$ .

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

Step 12.2

Reorder factors in $\frac{3 (x + 5)}{2 (x - 5) x^{2}}$ .

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