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

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

Evaluate

$$\frac{-2x^{3}+13x^{2}+8x-76}{3\left(x-4\right)\left(x-2\right)^{2}}$$

Short Solution Steps

Factor 3x^{2}-18x+24. Factor x^{2}-4x+4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(x-4\right)\left(x-2\right) and \left(x-2\right)^{2} is 3\left(x-4\right)\left(x-2\right)^{2}. Multiply \frac{x^{2}+2}{3\left(x-4\right)\left(x-2\right)} times \frac{x-2}{x-2}. Multiply \frac{x^{2}-x-6}{\left(x-2\right)^{2}} times \frac{3\left(x-4\right)}{3\left(x-4\right)}.

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

Since \frac{\left(x^{2}+2\right)\left(x-2\right)}{3\left(x-4\right)\left(x-2\right)^{2}} and \frac{\left(x^{2}-x-6\right)\times 3\left(x-4\right)}{3\left(x-4\right)\left(x-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{2}+2\right)\left(x-2\right)-\left(x^{2}-x-6\right)\times 3\left(x-4\right).

$$\frac{x^{3}-2x^{2}+2x-4-3x^{3}+12x^{2}+3x^{2}-12x+18x-72}{3\left(x-4\right)\left(x-2\right)^{2}}$$

Combine like terms in x^{3}-2x^{2}+2x-4-3x^{3}+12x^{2}+3x^{2}-12x+18x-72.

$$\frac{-2x^{3}+13x^{2}+8x-76}{3\left(x-4\right)\left(x-2\right)^{2}}$$

Expand 3\left(x-4\right)\left(x-2\right)^{2}.

$$\frac{-2x^{3}+13x^{2}+8x-76}{3x^{3}-24x^{2}+60x-48}$$

Expand

$$-\frac{2x^{3}-13x^{2}-8x+76}{3\left(x-4\right)\left(x-2\right)^{2}}$$

Short Solution Steps

Factor 3x^{2}-18x+24. Factor x^{2}-4x+4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(x-4\right)\left(x-2\right) and \left(x-2\right)^{2} is 3\left(x-4\right)\left(x-2\right)^{2}. Multiply \frac{x^{2}+2}{3\left(x-4\right)\left(x-2\right)} times \frac{x-2}{x-2}. Multiply \frac{x^{2}-x-6}{\left(x-2\right)^{2}} times \frac{3\left(x-4\right)}{3\left(x-4\right)}.

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

Since \frac{\left(x^{2}+2\right)\left(x-2\right)}{3\left(x-4\right)\left(x-2\right)^{2}} and \frac{\left(x^{2}-x-6\right)\times 3\left(x-4\right)}{3\left(x-4\right)\left(x-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{2}+2\right)\left(x-2\right)-\left(x^{2}-x-6\right)\times 3\left(x-4\right).

$$\frac{x^{3}-2x^{2}+2x-4-3x^{3}+12x^{2}+3x^{2}-12x+18x-72}{3\left(x-4\right)\left(x-2\right)^{2}}$$

Combine like terms in x^{3}-2x^{2}+2x-4-3x^{3}+12x^{2}+3x^{2}-12x+18x-72.

$$\frac{-2x^{3}+13x^{2}+8x-76}{3\left(x-4\right)\left(x-2\right)^{2}}$$

Expand 3\left(x-4\right)\left(x-2\right)^{2}.

$$\frac{-2x^{3}+13x^{2}+8x-76}{3x^{3}-24x^{2}+60x-48}$$