Simplify (1-1/(x^2))/((x+6)/5+1/x)

$\frac{1 - \frac{1}{x^{2}}}{\frac{x + 6}{5} + \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $5 x^{2}$ .

Multiply $\frac{1 - \frac{1}{x^{2}}}{\frac{x + 6}{5} + \frac{1}{x}}$ by $\frac{5 x^{2}}{5 x^{2}}$ .

$\frac{5 x^{2}}{5 x^{2}} \cdot \frac{1 - \frac{1}{x^{2}}}{\frac{x + 6}{5} + \frac{1}{x}}$

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

$\frac{5 x^{2} \cdot 1 + 5 x^{2} \frac{- 1}{x^{2}}}{5 x^{2} \frac{x + 6}{5} + 5 x^{2} \frac{1}{x}}$

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

$\frac{5 x^{2} \cdot 1 + x^{2} \cdot 5 \frac{- 1}{x^{2}}}{5 x^{2} \frac{x + 6}{5} + 5 x^{2} \frac{1}{x}}$

Cancel the common factor.

$\frac{5 x^{2} \cdot 1 + x^{2} \cdot 5 \frac{- 1}{x^{2}}}{5 x^{2} \frac{x + 6}{5} + 5 x^{2} \frac{1}{x}}$

Rewrite the expression.

$\frac{5 x^{2} \cdot 1 + 5 \cdot - 1}{5 x^{2} \frac{x + 6}{5} + 5 x^{2} \frac{1}{x}}$

Multiply $5$ by $- 1$ .

$\frac{5 x^{2} \cdot 1 - 5}{5 x^{2} \frac{x + 6}{5} + 5 x^{2} \frac{1}{x}}$

Cancel the common factor of $5$ .

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

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

Cancel the common factor.

$\frac{5 x^{2} \cdot 1 - 5}{5 x^{2} \frac{x + 6}{5} + 5 x^{2} \frac{1}{x}}$

Rewrite the expression.

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

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

Factor $5$ out of $5 x^{2} \cdot 1 - 5$ .

Factor $5$ out of $5 x^{2} \cdot 1$ .

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

Factor $5$ out of $- 5$ .

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

Factor $5$ out of $5 (x^{2} \cdot 1) + 5 (- 1)$ .

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

Rewrite $x^{2} \cdot 1$ as ${(x \cdot 1)}^{2}$ .

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

Rewrite $1$ as $1^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x \cdot 1$ and $b = 1$ .

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

Simplify.

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

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

Step 5

Simplify the denominator.

Factor $x$ out of $x^{2} (x + 6) + 5 x$ .

Factor $x$ out of $x^{2} (x + 6)$ .

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

Factor $x$ out of $5 x$ .

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

Factor $x$ out of $x (x (x + 6)) + x \cdot 5$ .

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

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Move $6$ to the left of $x$ .

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

Factor $x^{2} + 6 x + 5$ using the AC method.

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 $5$ and whose sum is $6$ .

$1, 5$

Write the factored form using these integers.

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

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

Step 6

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (1-1/(x^2))/((x+6)/5+1/x)

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