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

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Move the leading negative in $- \frac{4}{x^{2}}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

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

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

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

Factor $x$ out of $x \cdot 3$ .

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

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

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

Move $2$ to the left of $x$ .

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

Step 5

Simplify the denominator.

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

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

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

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

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

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

Simplify.

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

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

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