Solve 2x-x^2/x^2-6x+9+x^2-12x+20/x^2+6x-27

$$\frac { 2 x - x ^ { 2 } } { x ^ { 2 } - 6 x + 9 } + \frac { x ^ { 2 } - 12 x + 20 } { x ^ { 2 } + 6 x - 27 }$$

Evaluate

$$\frac{2\left(2-x\right)\left(11x-15\right)}{\left(x+9\right)\left(x-3\right)^{2}}$$

Short Solution Steps

Factor x^{2}-6x+9. Factor x^{2}+6x-27.

$$\frac{2x-x^{2}}{\left(x-3\right)^{2}}+\frac{x^{2}-12x+20}{\left(x-3\right)\left(x+9\right)}$$

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(x+9\right) is \left(x+9\right)\left(x-3\right)^{2}. Multiply \frac{2x-x^{2}}{\left(x-3\right)^{2}} times \frac{x+9}{x+9}. Multiply \frac{x^{2}-12x+20}{\left(x-3\right)\left(x+9\right)} times \frac{x-3}{x-3}.

$$\frac{\left(2x-x^{2}\right)\left(x+9\right)}{\left(x+9\right)\left(x-3\right)^{2}}+\frac{\left(x^{2}-12x+20\right)\left(x-3\right)}{\left(x+9\right)\left(x-3\right)^{2}}$$

Since \frac{\left(2x-x^{2}\right)\left(x+9\right)}{\left(x+9\right)\left(x-3\right)^{2}} and \frac{\left(x^{2}-12x+20\right)\left(x-3\right)}{\left(x+9\right)\left(x-3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(2x-x^{2}\right)\left(x+9\right)+\left(x^{2}-12x+20\right)\left(x-3\right)}{\left(x+9\right)\left(x-3\right)^{2}}$$

Do the multiplications in \left(2x-x^{2}\right)\left(x+9\right)+\left(x^{2}-12x+20\right)\left(x-3\right).

$$\frac{2x^{2}+18x-x^{3}-9x^{2}+x^{3}-3x^{2}-12x^{2}+36x+20x-60}{\left(x+9\right)\left(x-3\right)^{2}}$$

Combine like terms in 2x^{2}+18x-x^{3}-9x^{2}+x^{3}-3x^{2}-12x^{2}+36x+20x-60.

$$\frac{-22x^{2}+74x-60}{\left(x+9\right)\left(x-3\right)^{2}}$$

Expand \left(x+9\right)\left(x-3\right)^{2}.

$$\frac{-22x^{2}+74x-60}{x^{3}+3x^{2}-45x+81}$$

Expand

$$-\frac{2\left(11x^{2}-37x+30\right)}{\left(x+9\right)\left(x-3\right)^{2}}$$

Short Solution Steps

Factor x^{2}-6x+9. Factor x^{2}+6x-27.

$$\frac{2x-x^{2}}{\left(x-3\right)^{2}}+\frac{x^{2}-12x+20}{\left(x-3\right)\left(x+9\right)}$$

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(x+9\right) is \left(x+9\right)\left(x-3\right)^{2}. Multiply \frac{2x-x^{2}}{\left(x-3\right)^{2}} times \frac{x+9}{x+9}. Multiply \frac{x^{2}-12x+20}{\left(x-3\right)\left(x+9\right)} times \frac{x-3}{x-3}.

$$\frac{\left(2x-x^{2}\right)\left(x+9\right)}{\left(x+9\right)\left(x-3\right)^{2}}+\frac{\left(x^{2}-12x+20\right)\left(x-3\right)}{\left(x+9\right)\left(x-3\right)^{2}}$$

Since \frac{\left(2x-x^{2}\right)\left(x+9\right)}{\left(x+9\right)\left(x-3\right)^{2}} and \frac{\left(x^{2}-12x+20\right)\left(x-3\right)}{\left(x+9\right)\left(x-3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(2x-x^{2}\right)\left(x+9\right)+\left(x^{2}-12x+20\right)\left(x-3\right)}{\left(x+9\right)\left(x-3\right)^{2}}$$

Do the multiplications in \left(2x-x^{2}\right)\left(x+9\right)+\left(x^{2}-12x+20\right)\left(x-3\right).

$$\frac{2x^{2}+18x-x^{3}-9x^{2}+x^{3}-3x^{2}-12x^{2}+36x+20x-60}{\left(x+9\right)\left(x-3\right)^{2}}$$

Combine like terms in 2x^{2}+18x-x^{3}-9x^{2}+x^{3}-3x^{2}-12x^{2}+36x+20x-60.

$$\frac{-22x^{2}+74x-60}{\left(x+9\right)\left(x-3\right)^{2}}$$

Expand \left(x+9\right)\left(x-3\right)^{2}.

$$\frac{-22x^{2}+74x-60}{x^{3}+3x^{2}-45x+81}$$