Solve 6-2x/x^2-6x+9+(1/x-3-1/x+3)

$$\frac{ 6-2x }{ { x }^{ 2 } -6x+9 } +( \frac{ 1 }{ x-3 } - \frac{ 1 }{ x+3 } )$$

Evaluate

$$-\frac{2x}{x^{2}-9}$$

Short Solution Steps

Factor x^{2}-6x+9.

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine like terms in 3x+9-x^{2}-3x-x^{2}+6x-9.

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

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

$$\frac{6x-2x^{2}}{x^{3}-3x^{2}-9x+27}$$

Expand

$$-\frac{2x}{x^{2}-9}$$

Short Solution Steps

Factor x^{2}-6x+9.

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine like terms in 3x+9-x^{2}-3x-x^{2}+6x-9.

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

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

$$\frac{6x-2x^{2}}{x^{3}-3x^{2}-9x+27}$$