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

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

Step 1

Multiply the numerator and denominator of the complex fraction by $6 x^{2}$ .

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $2$ .

Factor $2$ out of $6 x^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Factor $x^{2}$ out of $6 x^{2}$ .

$\frac{3 x^{2} + x^{2} \cdot 6 \frac{- 2}{x^{2}}}{6 x^{2} \frac{x - 5}{6} + 6 x^{2} \frac{1}{x}}$

Cancel the common factor.

$\frac{3 x^{2} + x^{2} \cdot 6 \frac{- 2}{x^{2}}}{6 x^{2} \frac{x - 5}{6} + 6 x^{2} \frac{1}{x}}$

Rewrite the expression.

$\frac{3 x^{2} + 6 \cdot - 2}{6 x^{2} \frac{x - 5}{6} + 6 x^{2} \frac{1}{x}}$

Multiply $6$ by $- 2$ .

$\frac{3 x^{2} - 12}{6 x^{2} \frac{x - 5}{6} + 6 x^{2} \frac{1}{x}}$

Cancel the common factor of $6$ .

Factor $6$ out of $6 x^{2}$ .

$\frac{3 x^{2} - 12}{6 (x^{2}) \frac{x - 5}{6} + 6 x^{2} \frac{1}{x}}$

Cancel the common factor.

$\frac{3 x^{2} - 12}{6 x^{2} \frac{x - 5}{6} + 6 x^{2} \frac{1}{x}}$

Rewrite the expression.

$\frac{3 x^{2} - 12}{x^{2} (x - 5) + 6 x^{2} \frac{1}{x}}$

Cancel the common factor of $x$ .

Factor $x$ out of $6 x^{2}$ .

$\frac{3 x^{2} - 12}{x^{2} (x - 5) + x (6 x) \frac{1}{x}}$

Cancel the common factor.

$\frac{3 x^{2} - 12}{x^{2} (x - 5) + x (6 x) \frac{1}{x}}$

Rewrite the expression.

$\frac{3 x^{2} - 12}{x^{2} (x - 5) + 6 x}$

Step 4

Simplify the numerator.

Factor $3$ out of $3 x^{2} - 12$ .

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) - 12}{x^{2} (x - 5) + 6 x}$

Factor $3$ out of $- 12$ .

$\frac{3 x^{2} + 3 \cdot - 4}{x^{2} (x - 5) + 6 x}$

Factor $3$ out of $3 x^{2} + 3 \cdot - 4$ .

$\frac{3 (x^{2} - 4)}{x^{2} (x - 5) + 6 x}$

Rewrite $4$ as $2^{2}$ .

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

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

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

Step 5

Simplify the denominator.

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

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

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

Factor $x$ out of $6 x$ .

$\frac{3 (x + 2) (x - 2)}{x (x (x - 5)) + x \cdot 6}$

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

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

Apply the distributive property.

$\frac{3 (x + 2) (x - 2)}{x (x \cdot x + x \cdot - 5 + 6)}$

Multiply $x$ by $x$ .

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

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

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

Factor $x^{2} - 5 x + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

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

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

Step 6

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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