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

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

Evaluate

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

Short Solution Steps

Factor x^{2}-4.

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

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

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

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

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

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

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

Combine like terms in 3x-6+x+2.

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

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

$$\frac{4x-4}{x^{3}-2x^{2}-4x+8}$$

Expand

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

Short Solution Steps

Factor x^{2}-4.

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

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

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

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

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

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

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

Combine like terms in 3x-6+x+2.

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

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

$$\frac{4x-4}{x^{3}-2x^{2}-4x+8}$$