Solve [x/(x-3)^2-6x/(x-3)^2*(x+3)]*2x*(x-3)(x+3)/2x^2

$$[ \frac { x } { ( x - 3 ) ^ { 2 } } - \frac { 6 x } { ( x - 3 ) ^ { 2 } \cdot ( x + 3 ) } ] \cdot \frac { 2 x \cdot ( x - 3 ) ( x + 3 ) } { 2 x ^ { 2 } }$$

Evaluate

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Short Solution Steps

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

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

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

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

Do the multiplications in x\left(x+3\right)-6x.

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

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

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

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

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

Cancel out x-3 in both numerator and denominator.

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

Cancel out 2x in both numerator and denominator.

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

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

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

Cancel out x\left(x-3\right)\left(x+3\right) in both numerator and denominator.

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Factor

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