Algebra Examples

$\frac{1}{9 x^{2} - 25} - \frac{1}{6 x + 10}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\frac{1}{{(3 x)}^{2} - 25} - \frac{1}{6 x + 10}$

Step 1.1.2

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

$\frac{1}{{(3 x)}^{2} - 5^{2}} - \frac{1}{6 x + 10}$

Step 1.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 5$ .

$\frac{1}{(3 x + 5) (3 x - 5)} - \frac{1}{6 x + 10}$

Step 1.2

Factor $2$ out of $6 x + 10$ .

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Step 1.2.1

Factor $2$ out of $6 x$ .

$\frac{1}{(3 x + 5) (3 x - 5)} - \frac{1}{2 (3 x) + 10}$

Step 1.2.2

Factor $2$ out of $10$ .

$\frac{1}{(3 x + 5) (3 x - 5)} - \frac{1}{2 (3 x) + 2 (5)}$

Step 1.2.3

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

$\frac{1}{(3 x + 5) (3 x - 5)} - \frac{1}{2 (3 x + 5)}$

Step 2

To write $\frac{1}{(3 x + 5) (3 x - 5)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

To write $- \frac{1}{2 (3 x + 5)}$ as a fraction with a common denominator, multiply by $\frac{3 x - 5}{3 x - 5}$ .

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

Step 4

Write each expression with a common denominator of $(3 x + 5) (3 x - 5) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{(3 x + 5) (3 x - 5)}$ by $\frac{2}{2}$ .

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

Step 4.2

Multiply $\frac{1}{2 (3 x + 5)}$ by $\frac{3 x - 5}{3 x - 5}$ .

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

Step 4.3

Reorder the factors of $(3 x + 5) (3 x - 5) \cdot 2$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $3$ by $- 1$ .

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

Step 6.3

Multiply $- 1$ by $- 5$ .

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

Step 6.4

Add $2$ and $5$ .

$\frac{- 3 x + 7}{2 (3 x + 5) (3 x - 5)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- 3 x$ .

$\frac{- (3 x) + 7}{2 (3 x + 5) (3 x - 5)}$

Step 7.2

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{- (3 x) - 1 (- 7)}{2 (3 x + 5) (3 x - 5)}$

Step 7.3

Factor $- 1$ out of $- (3 x) - 1 (- 7)$ .

$\frac{- (3 x - 7)}{2 (3 x + 5) (3 x - 5)}$

Step 7.4

Simplify the expression.

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Step 7.4.1

Rewrite $- (3 x - 7)$ as $- 1 (3 x - 7)$ .

$\frac{- 1 (3 x - 7)}{2 (3 x + 5) (3 x - 5)}$

Step 7.4.2

Move the negative in front of the fraction.

$- \frac{3 x - 7}{2 (3 x + 5) (3 x - 5)}$