Algebra Examples

$\frac{6 x - 1}{x^{2} - 25} - \frac{7 x - 6}{x^{2} - 25}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

$\frac{6 x - 1 - (7 x - 6)}{x^{2} - 25}$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$\frac{6 x - 1 - (7 x) - - 6}{x^{2} - 25}$

Step 1.2.2

Multiply $7$ by $- 1$ .

$\frac{6 x - 1 - 7 x - - 6}{x^{2} - 25}$

Step 1.2.3

Multiply $- 1$ by $- 6$ .

$\frac{6 x - 1 - 7 x + 6}{x^{2} - 25}$

Step 1.3

Simplify by adding terms.

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Step 1.3.1

Subtract $7 x$ from $6 x$ .

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

Step 1.3.2

Add $- 1$ and $6$ .

$\frac{- x + 5}{x^{2} - 25}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{- x + 5}{x^{2} - 5^{2}}$

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 $b = 5$ .

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Cancel the common factor.

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

Step 3.1.6

Rewrite the expression.

$\frac{- 1}{x + 5}$

Step 3.2

Move the negative in front of the fraction.

$- \frac{1}{x + 5}$