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

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Move the leading negative in $- \frac{3}{x^{2}}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Factor $x^{2} \cdot 1 + x - 2$ 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 $1$ .

$- 1, 2$

Write the factored form using these integers.

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

Step 5

Factor $x^{2} \cdot 1 + x \cdot 2 - 3$ 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Write the factored form using these integers.

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

Step 6

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 2}{x + 3}$

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

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