Solve x^2+2/3x^2-18x+24divx^2-x-6/x^2-4x+4

$$\frac { x ^ { 2 } + 2 } { 3 x ^ { 2 } - 18 x + 24 } \div \frac { x ^ { 2 } - x - 6 } { x ^ { 2 } - 4 x + 4 }$$

Evaluate

$$\frac{\left(x-2\right)\left(x^{2}+2\right)}{3\left(x-4\right)\left(x-3\right)\left(x+2\right)}$$

Short Solution Steps

Divide \frac{x^{2}+2}{3x^{2}-18x+24} by \frac{x^{2}-x-6}{x^{2}-4x+4} by multiplying \frac{x^{2}+2}{3x^{2}-18x+24} by the reciprocal of \frac{x^{2}-x-6}{x^{2}-4x+4}.

$$\frac{\left(x^{2}+2\right)\left(x^{2}-4x+4\right)}{\left(3x^{2}-18x+24\right)\left(x^{2}-x-6\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(x-2\right)^{2}\left(x^{2}+2\right)}{3\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+2\right)}$$

Cancel out x-2 in both numerator and denominator.

$$\frac{\left(x-2\right)\left(x^{2}+2\right)}{3\left(x-4\right)\left(x-3\right)\left(x+2\right)}$$

Expand the expression.

$$\frac{x^{3}-2x^{2}+2x-4}{3x^{3}-15x^{2}-6x+72}$$

Expand

$$\frac{x^{3}-2x^{2}+2x-4}{3\left(x-4\right)\left(x^{2}-x-6\right)}$$

Short Solution Steps

Divide \frac{x^{2}+2}{3x^{2}-18x+24} by \frac{x^{2}-x-6}{x^{2}-4x+4} by multiplying \frac{x^{2}+2}{3x^{2}-18x+24} by the reciprocal of \frac{x^{2}-x-6}{x^{2}-4x+4}.

$$\frac{\left(x^{2}+2\right)\left(x^{2}-4x+4\right)}{\left(3x^{2}-18x+24\right)\left(x^{2}-x-6\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(x-2\right)^{2}\left(x^{2}+2\right)}{3\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+2\right)}$$

Cancel out x-2 in both numerator and denominator.

$$\frac{\left(x-2\right)\left(x^{2}+2\right)}{3\left(x-4\right)\left(x-3\right)\left(x+2\right)}$$

Expand the expression.

$$\frac{x^{3}-2x^{2}+2x-4}{3x^{3}-15x^{2}-6x+72}$$