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

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

Evaluate

$$\frac{3x-8}{\left(x-2\right)^{3}}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{2x-4}{\left(x-2\right)^{4}}.

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

Cancel out x-2 in both numerator and denominator.

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

Express \left(x-3\right)\times \frac{2}{\left(x-2\right)^{3}} as a single fraction.

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

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)^{3} is \left(x-2\right)^{3}. Multiply \frac{1}{\left(x-2\right)^{2}} times \frac{x-2}{x-2}.

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

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

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

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

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

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

$$\frac{3x-8}{\left(x-2\right)^{3}}$$

Expand \left(x-2\right)^{3}.

$$\frac{3x-8}{x^{3}-6x^{2}+12x-8}$$

Expand

$$\frac{3x-8}{\left(x-2\right)^{3}}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{2x-4}{\left(x-2\right)^{4}}.

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

Cancel out x-2 in both numerator and denominator.

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

Express \left(x-3\right)\times \frac{2}{\left(x-2\right)^{3}} as a single fraction.

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

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)^{3} is \left(x-2\right)^{3}. Multiply \frac{1}{\left(x-2\right)^{2}} times \frac{x-2}{x-2}.

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

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

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

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

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

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

$$\frac{3x-8}{\left(x-2\right)^{3}}$$

Expand \left(x-2\right)^{3}.

$$\frac{3x-8}{x^{3}-6x^{2}+12x-8}$$