Solve 6+2x/x^{2-5x+6}:(x^2+6x+9/x^2-2x:9-x^2/x^2)

$$\frac { 6 + 2 x } { x ^ { 2 } - 5 x + 6 } : ( \frac { x ^ { 2 } + 6 x + 9 } { x ^ { 2 } - 2 x } : \frac { 9 - x ^ { 2 } } { x ^ { 2 } } )$$

Evaluate

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

Short Solution Steps

Divide \frac{6+2x}{x^{2}-5x+6} by \frac{\frac{x^{2}+6x+9}{x^{2}-2x}}{\frac{9-x^{2}}{x^{2}}} by multiplying \frac{6+2x}{x^{2}-5x+6} by the reciprocal of \frac{\frac{x^{2}+6x+9}{x^{2}-2x}}{\frac{9-x^{2}}{x^{2}}}.

$$\frac{\left(6+2x\right)\times \frac{9-x^{2}}{x^{2}}}{\left(x^{2}-5x+6\right)\times \frac{x^{2}+6x+9}{x^{2}-2x}}$$

Express \left(6+2x\right)\times \frac{9-x^{2}}{x^{2}} as a single fraction.

$$\frac{\frac{\left(6+2x\right)\left(9-x^{2}\right)}{x^{2}}}{\left(x^{2}-5x+6\right)\times \frac{x^{2}+6x+9}{x^{2}-2x}}$$

Express \left(x^{2}-5x+6\right)\times \frac{x^{2}+6x+9}{x^{2}-2x} as a single fraction.

$$\frac{\frac{\left(6+2x\right)\left(9-x^{2}\right)}{x^{2}}}{\frac{\left(x^{2}-5x+6\right)\left(x^{2}+6x+9\right)}{x^{2}-2x}}$$

Divide \frac{\left(6+2x\right)\left(9-x^{2}\right)}{x^{2}} by \frac{\left(x^{2}-5x+6\right)\left(x^{2}+6x+9\right)}{x^{2}-2x} by multiplying \frac{\left(6+2x\right)\left(9-x^{2}\right)}{x^{2}} by the reciprocal of \frac{\left(x^{2}-5x+6\right)\left(x^{2}+6x+9\right)}{x^{2}-2x}.

$$\frac{\left(6+2x\right)\left(9-x^{2}\right)\left(x^{2}-2x\right)}{x^{2}\left(x^{2}-5x+6\right)\left(x^{2}+6x+9\right)}$$

Factor the expressions that are not already factored.

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

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

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

Extract the negative sign in -3-x.

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

Cancel out x+3 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Divide \frac{6+2x}{x^{2}-5x+6} by \frac{\frac{x^{2}+6x+9}{x^{2}-2x}}{\frac{9-x^{2}}{x^{2}}} by multiplying \frac{6+2x}{x^{2}-5x+6} by the reciprocal of \frac{\frac{x^{2}+6x+9}{x^{2}-2x}}{\frac{9-x^{2}}{x^{2}}}.

$$\frac{\left(6+2x\right)\times \frac{9-x^{2}}{x^{2}}}{\left(x^{2}-5x+6\right)\times \frac{x^{2}+6x+9}{x^{2}-2x}}$$

Express \left(6+2x\right)\times \frac{9-x^{2}}{x^{2}} as a single fraction.

$$\frac{\frac{\left(6+2x\right)\left(9-x^{2}\right)}{x^{2}}}{\left(x^{2}-5x+6\right)\times \frac{x^{2}+6x+9}{x^{2}-2x}}$$

Express \left(x^{2}-5x+6\right)\times \frac{x^{2}+6x+9}{x^{2}-2x} as a single fraction.

$$\frac{\frac{\left(6+2x\right)\left(9-x^{2}\right)}{x^{2}}}{\frac{\left(x^{2}-5x+6\right)\left(x^{2}+6x+9\right)}{x^{2}-2x}}$$

Divide \frac{\left(6+2x\right)\left(9-x^{2}\right)}{x^{2}} by \frac{\left(x^{2}-5x+6\right)\left(x^{2}+6x+9\right)}{x^{2}-2x} by multiplying \frac{\left(6+2x\right)\left(9-x^{2}\right)}{x^{2}} by the reciprocal of \frac{\left(x^{2}-5x+6\right)\left(x^{2}+6x+9\right)}{x^{2}-2x}.

$$\frac{\left(6+2x\right)\left(9-x^{2}\right)\left(x^{2}-2x\right)}{x^{2}\left(x^{2}-5x+6\right)\left(x^{2}+6x+9\right)}$$

Factor the expressions that are not already factored.

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

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

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

Extract the negative sign in -3-x.

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

Cancel out x+3 in both numerator and denominator.

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