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

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

Evaluate

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Short Solution Steps

To raise \frac{x-1}{x-3} to a power, raise both numerator and denominator to the power and then divide.

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

Express 2\times \frac{x-1}{x-3} as a single fraction.

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

Multiply \frac{2\left(x-1\right)}{x-3} times \frac{2}{3-x} by multiplying numerator times numerator and denominator times denominator.

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

To raise \frac{2}{3-x} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Since \frac{\left(x-1\right)^{2}}{\left(x-3\right)^{2}} and \frac{-2\left(x-1\right)\times 2}{\left(x-3\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x-1\right)^{2}-2\left(x-1\right)\times 2.

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

Combine like terms in x^{2}-2x+1-4x+4.

$$\frac{x^{2}-6x+5}{\left(x-3\right)^{2}}+\frac{2^{2}}{\left(3-x\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand \left(3-x\right)^{2}.

$$\frac{x^{2}-6x+5}{\left(x-3\right)^{2}}+\frac{2^{2}}{\left(x-3\right)^{2}}$$

Since \frac{x^{2}-6x+5}{\left(x-3\right)^{2}} and \frac{2^{2}}{\left(x-3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}-6x+5+2^{2}}{\left(x-3\right)^{2}}$$

Combine like terms in x^{2}-6x+5+2^{2}.

$$\frac{x^{2}-6x+9}{\left(x-3\right)^{2}}$$

Factor the expressions that are not already factored in \frac{x^{2}-6x+9}{\left(x-3\right)^{2}}.

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

Cancel out \left(x-3\right)^{2} in both numerator and denominator.

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Factor

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