Solve x^2-6x+9/9-x^2-2x-6/x^2+3x

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

Evaluate

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

Short Solution Steps

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

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

Cancel out x-3 in both numerator and denominator.

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

Factor x^{2}+3x.

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

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

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

Since \frac{\left(x-3\right)\left(-1\right)x}{x\left(x+3\right)} and \frac{2x-6}{x\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-3\right)\left(-1\right)x-\left(2x-6\right).

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

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

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

Expand x\left(x+3\right).

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

Expand

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

Short Solution Steps

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

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

Cancel out x-3 in both numerator and denominator.

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

Factor x^{2}+3x.

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

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

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

Since \frac{\left(x-3\right)\left(-1\right)x}{x\left(x+3\right)} and \frac{2x-6}{x\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-3\right)\left(-1\right)x-\left(2x-6\right).

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

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

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

Expand x\left(x+3\right).

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