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

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

Evaluate

$$4\times \left(\frac{x}{x^{2}-4}\right)^{2}$$

Short Solution Steps

Factor x^{2}-4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)^{2} and \left(x-2\right)\left(x+2\right) is \left(x+2\right)\left(x-2\right)^{2}. Multiply \frac{1}{\left(x-2\right)^{2}} times \frac{x+2}{x+2}. Multiply \frac{2}{\left(x-2\right)\left(x+2\right)} times \frac{x-2}{x-2}.

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

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

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

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

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

Combine like terms in x+2+2x-4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)\left(x-2\right)^{2} and \left(x+2\right)^{2} is \left(x-2\right)^{2}\left(x+2\right)^{2}. Multiply \frac{3x-2}{\left(x+2\right)\left(x-2\right)^{2}} times \frac{x+2}{x+2}. Multiply \frac{1}{\left(x+2\right)^{2}} times \frac{\left(x-2\right)^{2}}{\left(x-2\right)^{2}}.

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

Since \frac{\left(3x-2\right)\left(x+2\right)}{\left(x-2\right)^{2}\left(x+2\right)^{2}} and \frac{\left(x-2\right)^{2}}{\left(x-2\right)^{2}\left(x+2\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(3x-2\right)\left(x+2\right)+\left(x-2\right)^{2}.

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

Combine like terms in 3x^{2}+6x-2x-4+x^{2}-4x+4.

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

Expand \left(x-2\right)^{2}\left(x+2\right)^{2}.

$$\frac{4x^{2}}{x^{4}-8x^{2}+16}$$

Expand

$$4\times \left(\frac{x}{x^{2}-4}\right)^{2}$$

Short Solution Steps

Factor x^{2}-4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)^{2} and \left(x-2\right)\left(x+2\right) is \left(x+2\right)\left(x-2\right)^{2}. Multiply \frac{1}{\left(x-2\right)^{2}} times \frac{x+2}{x+2}. Multiply \frac{2}{\left(x-2\right)\left(x+2\right)} times \frac{x-2}{x-2}.

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

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

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

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

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

Combine like terms in x+2+2x-4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)\left(x-2\right)^{2} and \left(x+2\right)^{2} is \left(x-2\right)^{2}\left(x+2\right)^{2}. Multiply \frac{3x-2}{\left(x+2\right)\left(x-2\right)^{2}} times \frac{x+2}{x+2}. Multiply \frac{1}{\left(x+2\right)^{2}} times \frac{\left(x-2\right)^{2}}{\left(x-2\right)^{2}}.

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

Since \frac{\left(3x-2\right)\left(x+2\right)}{\left(x-2\right)^{2}\left(x+2\right)^{2}} and \frac{\left(x-2\right)^{2}}{\left(x-2\right)^{2}\left(x+2\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(3x-2\right)\left(x+2\right)+\left(x-2\right)^{2}.

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

Combine like terms in 3x^{2}+6x-2x-4+x^{2}-4x+4.

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

Expand \left(x-2\right)^{2}\left(x+2\right)^{2}.

$$\frac{4x^{2}}{x^{4}-8x^{2}+16}$$