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

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-4 and x-2 is \left(x-4\right)\left(x-2\right). Multiply \frac{x-3}{x-4} times \frac{x-2}{x-2}. Multiply \frac{x-1}{x-2} times \frac{x-4}{x-4}.

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

Since \frac{\left(x-3\right)\left(x-2\right)}{\left(x-4\right)\left(x-2\right)} and \frac{\left(x-1\right)\left(x-4\right)}{\left(x-4\right)\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-3\right)\left(x-2\right)-\left(x-1\right)\left(x-4\right).

$$\frac{x^{2}-2x-3x+6-x^{2}+4x+x-4}{\left(x-4\right)\left(x-2\right)}-\frac{1}{\left(x-2\right)^{2}}$$

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4\right)\left(x-2\right) and \left(x-2\right)^{2} is \left(x-4\right)\left(x-2\right)^{2}. Multiply \frac{2}{\left(x-4\right)\left(x-2\right)} times \frac{x-2}{x-2}. Multiply \frac{1}{\left(x-2\right)^{2}} times \frac{x-4}{x-4}.

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

Since \frac{2\left(x-2\right)}{\left(x-4\right)\left(x-2\right)^{2}} and \frac{x-4}{\left(x-4\right)\left(x-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(x-2\right)-\left(x-4\right).

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

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

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

Expand \left(x-4\right)\left(x-2\right)^{2}.

$$\frac{x}{x^{3}-8x^{2}+20x-16}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-4 and x-2 is \left(x-4\right)\left(x-2\right). Multiply \frac{x-3}{x-4} times \frac{x-2}{x-2}. Multiply \frac{x-1}{x-2} times \frac{x-4}{x-4}.

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

Since \frac{\left(x-3\right)\left(x-2\right)}{\left(x-4\right)\left(x-2\right)} and \frac{\left(x-1\right)\left(x-4\right)}{\left(x-4\right)\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-3\right)\left(x-2\right)-\left(x-1\right)\left(x-4\right).

$$\frac{x^{2}-2x-3x+6-x^{2}+4x+x-4}{\left(x-4\right)\left(x-2\right)}-\frac{1}{\left(x-2\right)^{2}}$$

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4\right)\left(x-2\right) and \left(x-2\right)^{2} is \left(x-4\right)\left(x-2\right)^{2}. Multiply \frac{2}{\left(x-4\right)\left(x-2\right)} times \frac{x-2}{x-2}. Multiply \frac{1}{\left(x-2\right)^{2}} times \frac{x-4}{x-4}.

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

Since \frac{2\left(x-2\right)}{\left(x-4\right)\left(x-2\right)^{2}} and \frac{x-4}{\left(x-4\right)\left(x-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(x-2\right)-\left(x-4\right).

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

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

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

Expand \left(x-4\right)\left(x-2\right)^{2}.

$$\frac{x}{x^{3}-8x^{2}+20x-16}$$