Simplify (4/(x^2)-1)/((x+3)/2+1/x)

$\frac{\frac{4}{x^{2}} - 1}{\frac{x + 3}{2} + \frac{1}{x}}$

Step 1

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

Multiply $\frac{\frac{4}{x^{2}} - 1}{\frac{x + 3}{2} + \frac{1}{x}}$ by $\frac{2 x^{2}}{2 x^{2}}$ .

$\frac{2 x^{2}}{2 x^{2}} \cdot \frac{\frac{4}{x^{2}} - 1}{\frac{x + 3}{2} + \frac{1}{x}}$

Combine.

$\frac{2 x^{2} (\frac{4}{x^{2}} - 1)}{2 x^{2} (\frac{x + 3}{2} + \frac{1}{x})}$

Step 2

Apply the distributive property.

$\frac{2 x^{2} \frac{4}{x^{2}} + 2 x^{2} \cdot - 1}{2 x^{2} \frac{x + 3}{2} + 2 x^{2} \frac{1}{x}}$

Step 3

Simplify by cancelling.

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

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

$\frac{x^{2} \cdot 2 \frac{4}{x^{2}} + 2 x^{2} \cdot - 1}{2 x^{2} \frac{x + 3}{2} + 2 x^{2} \frac{1}{x}}$

Cancel the common factor.

$\frac{x^{2} \cdot 2 \frac{4}{x^{2}} + 2 x^{2} \cdot - 1}{2 x^{2} \frac{x + 3}{2} + 2 x^{2} \frac{1}{x}}$

Rewrite the expression.

$\frac{2 \cdot 4 + 2 x^{2} \cdot - 1}{2 x^{2} \frac{x + 3}{2} + 2 x^{2} \frac{1}{x}}$

Multiply $2$ by $4$ .

$\frac{8 + 2 x^{2} \cdot - 1}{2 x^{2} \frac{x + 3}{2} + 2 x^{2} \frac{1}{x}}$

Cancel the common factor of $2$ .

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

$\frac{8 + 2 x^{2} \cdot - 1}{2 (x^{2}) \frac{x + 3}{2} + 2 x^{2} \frac{1}{x}}$

Cancel the common factor.

$\frac{8 + 2 x^{2} \cdot - 1}{2 x^{2} \frac{x + 3}{2} + 2 x^{2} \frac{1}{x}}$

Rewrite the expression.

$\frac{8 + 2 x^{2} \cdot - 1}{x^{2} (x + 3) + 2 x^{2} \frac{1}{x}}$

Cancel the common factor of $x$ .

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

$\frac{8 + 2 x^{2} \cdot - 1}{x^{2} (x + 3) + x (2 x) \frac{1}{x}}$

Cancel the common factor.

$\frac{8 + 2 x^{2} \cdot - 1}{x^{2} (x + 3) + x (2 x) \frac{1}{x}}$

Rewrite the expression.

$\frac{8 + 2 x^{2} \cdot - 1}{x^{2} (x + 3) + 2 x}$

Step 4

Simplify the numerator.

Multiply $- 1$ by $2$ .

$\frac{8 - 2 x^{2}}{x^{2} (x + 3) + 2 x}$

Rewrite $8 - 2 x^{2}$ in a factored form.

Factor $2$ out of $8 - 2 x^{2}$ .

Factor $2$ out of $8$ .

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

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

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

Factor $2$ out of $2 (4) + 2 (- x^{2})$ .

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

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

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

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

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

Step 5

Simplify the denominator.

Factor $x$ out of $x^{2} (x + 3) + 2 x$ .

Factor $x$ out of $x^{2} (x + 3)$ .

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

Factor $x$ out of $2 x$ .

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

Factor $x$ out of $x (x (x + 3)) + x \cdot 2$ .

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

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Move $3$ to the left of $x$ .

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

Factor $x^{2} + 3 x + 2$ 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 $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

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

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

Step 6

Cancel the common factor of $2 + x$ and $x + 2$ .

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (4/(x^2)-1)/((x+3)/2+1/x)

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