Solve x^2-4x+4/x^2-4+x-2/x^2+2x-1

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-4x+4}{x^{2}-4}.

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

Cancel out x-2 in both numerator and denominator.

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

Factor x^{2}+2x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x\left(x+2\right) is x\left(x+2\right). Multiply \frac{x-2}{x+2} times \frac{x}{x}.

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

Since \frac{\left(x-2\right)x}{x\left(x+2\right)} and \frac{x-2}{x\left(x+2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x-2\right)x+x-2.

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

Combine like terms in x^{2}-2x+x-2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x\left(x+2\right)}{x\left(x+2\right)}.

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

Since \frac{x^{2}-x-2}{x\left(x+2\right)} and \frac{x\left(x+2\right)}{x\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}-x-2-x\left(x+2\right).

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

Combine like terms in x^{2}-x-2-x^{2}-2x.

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

Expand x\left(x+2\right).

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-4x+4}{x^{2}-4}.

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

Cancel out x-2 in both numerator and denominator.

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

Factor x^{2}+2x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x\left(x+2\right) is x\left(x+2\right). Multiply \frac{x-2}{x+2} times \frac{x}{x}.

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

Since \frac{\left(x-2\right)x}{x\left(x+2\right)} and \frac{x-2}{x\left(x+2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x-2\right)x+x-2.

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

Combine like terms in x^{2}-2x+x-2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x\left(x+2\right)}{x\left(x+2\right)}.

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

Since \frac{x^{2}-x-2}{x\left(x+2\right)} and \frac{x\left(x+2\right)}{x\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}-x-2-x\left(x+2\right).

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

Combine like terms in x^{2}-x-2-x^{2}-2x.

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

Expand x\left(x+2\right).

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