Algebra Examples

$\frac{4}{9 x^{2} - 45 x} - \frac{6 x}{x^{2} - 25}$

Step 1

Simplify each term.

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Step 1.1

Factor $9 x$ out of $9 x^{2} - 45 x$ .

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Step 1.1.1

Factor $9 x$ out of $9 x^{2}$ .

$\frac{4}{9 x (x) - 45 x} - \frac{6 x}{x^{2} - 25}$

Step 1.1.2

Factor $9 x$ out of $- 45 x$ .

$\frac{4}{9 x (x) + 9 x (- 5)} - \frac{6 x}{x^{2} - 25}$

Step 1.1.3

Factor $9 x$ out of $9 x (x) + 9 x (- 5)$ .

$\frac{4}{9 x (x - 5)} - \frac{6 x}{x^{2} - 25}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

$\frac{4}{9 x (x - 5)} - \frac{6 x}{x^{2} - 5^{2}}$

Step 1.2.2

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 $b = 5$ .

$\frac{4}{9 x (x - 5)} - \frac{6 x}{(x + 5) (x - 5)}$

Step 2

To write $\frac{4}{9 x (x - 5)}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{4}{9 x (x - 5)} \cdot \frac{x + 5}{x + 5} - \frac{6 x}{(x + 5) (x - 5)}$

Step 3

To write $- \frac{6 x}{(x + 5) (x - 5)}$ as a fraction with a common denominator, multiply by $\frac{9 x}{9 x}$ .

$\frac{4}{9 x (x - 5)} \cdot \frac{x + 5}{x + 5} - \frac{6 x}{(x + 5) (x - 5)} \cdot \frac{9 x}{9 x}$

Step 4

Write each expression with a common denominator of $9 x (x - 5) (x + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

$\frac{4 (x + 5)}{9 x (x - 5) (x + 5)} - \frac{6 x}{(x + 5) (x - 5)} \cdot \frac{9 x}{9 x}$

Step 4.2

Multiply $\frac{6 x}{(x + 5) (x - 5)}$ by $\frac{9 x}{9 x}$ .

$\frac{4 (x + 5)}{9 x (x - 5) (x + 5)} - \frac{6 x (9 x)}{(x + 5) (x - 5) (9 x)}$

Step 4.3

Reorder the factors of $9 x (x - 5) (x + 5)$ .

$\frac{4 (x + 5)}{9 x (x + 5) (x - 5)} - \frac{6 x (9 x)}{(x + 5) (x - 5) (9 x)}$

Step 4.4

Reorder the factors of $(x + 5) (x - 5) (9 x)$ .

$\frac{4 (x + 5)}{9 x (x + 5) (x - 5)} - \frac{6 x (9 x)}{9 x (x + 5) (x - 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{4 (x + 5) - 6 x (9 x)}{9 x (x + 5) (x - 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $2$ out of $4 (x + 5) - 6 x (9 x)$ .

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Step 6.1.1

Factor $2$ out of $4 (x + 5)$ .

$\frac{2 (2 (x + 5)) - 6 x (9 x)}{9 x (x + 5) (x - 5)}$

Step 6.1.2

Factor $2$ out of $- 6 x (9 x)$ .

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

Step 6.1.3

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $2$ by $5$ .

$\frac{2 (2 x + 10 - 3 x (9 x))}{9 x (x + 5) (x - 5)}$

Step 6.4

Rewrite using the commutative property of multiplication.

$\frac{2 (2 x + 10 - 3 \cdot 9 x \cdot x)}{9 x (x + 5) (x - 5)}$

Step 6.5

Multiply $x$ by $x$ by adding the exponents.

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Step 6.5.1

Move $x$ .

$\frac{2 (2 x + 10 - 3 \cdot 9 (x \cdot x))}{9 x (x + 5) (x - 5)}$

Step 6.5.2

Multiply $x$ by $x$ .

$\frac{2 (2 x + 10 - 3 \cdot 9 x^{2})}{9 x (x + 5) (x - 5)}$

Step 6.6

Multiply $- 3$ by $9$ .

$\frac{2 (2 x + 10 - 27 x^{2})}{9 x (x + 5) (x - 5)}$

Step 6.7

Reorder terms.

$\frac{2 (- 27 x^{2} + 2 x + 10)}{9 x (x + 5) (x - 5)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- 27 x^{2}$ .

$\frac{2 (- (27 x^{2}) + 2 x + 10)}{9 x (x + 5) (x - 5)}$

Step 7.2

Factor $- 1$ out of $2 x$ .

$\frac{2 (- (27 x^{2}) - (- 2 x) + 10)}{9 x (x + 5) (x - 5)}$

Step 7.3

Factor $- 1$ out of $- (27 x^{2}) - (- 2 x)$ .

$\frac{2 (- (27 x^{2} - 2 x) + 10)}{9 x (x + 5) (x - 5)}$

Step 7.4

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{2 (- (27 x^{2} - 2 x) - 1 (- 10))}{9 x (x + 5) (x - 5)}$

Step 7.5

Factor $- 1$ out of $- (27 x^{2} - 2 x) - 1 (- 10)$ .

$\frac{2 (- (27 x^{2} - 2 x - 10))}{9 x (x + 5) (x - 5)}$

Step 7.6

Simplify the expression.

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Step 7.6.1

Rewrite $- (27 x^{2} - 2 x - 10)$ as $- 1 (27 x^{2} - 2 x - 10)$ .

$\frac{2 (- 1 (27 x^{2} - 2 x - 10))}{9 x (x + 5) (x - 5)}$

Step 7.6.2

Move the negative in front of the fraction.

$- \frac{2 (27 x^{2} - 2 x - 10)}{9 x (x + 5) (x - 5)}$