Solve (3/x^2-4-2/x^2-4x+4)div(x-10/3x^2-12)=

$$( \frac { 3 } { x ^ { 2 } - 4 } - \frac { 2 } { x ^ { 2 } - 4 x + 4 } ) \div ( \frac { x - 10 } { 3 x ^ { 2 } - 12 } ) =$$

Evaluate

$$\frac{3}{x-2}$$

Short Solution Steps

Factor x^{2}-4. Factor x^{2}-4x+4.

$$\frac{\frac{3}{\left(x-2\right)\left(x+2\right)}-\frac{2}{\left(x-2\right)^{2}}}{\frac{x-10}{3x^{2}-12}}$$

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

$$\frac{\frac{3\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}}-\frac{2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{x-10}{3x^{2}-12}}$$

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

$$\frac{\frac{3\left(x-2\right)-2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{x-10}{3x^{2}-12}}$$

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

$$\frac{\frac{3x-6-2x-4}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{x-10}{3x^{2}-12}}$$

Combine like terms in 3x-6-2x-4.

$$\frac{\frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{x-10}{3x^{2}-12}}$$

Divide \frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}} by \frac{x-10}{3x^{2}-12} by multiplying \frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}} by the reciprocal of \frac{x-10}{3x^{2}-12}.

$$\frac{\left(x-10\right)\left(3x^{2}-12\right)}{\left(x+2\right)\left(x-2\right)^{2}\left(x-10\right)}$$

Cancel out x-10 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

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

$$\frac{3}{x-2}$$

Expand

$$\frac{3}{x-2}$$

Short Solution Steps

Factor x^{2}-4. Factor x^{2}-4x+4.

$$\frac{\frac{3}{\left(x-2\right)\left(x+2\right)}-\frac{2}{\left(x-2\right)^{2}}}{\frac{x-10}{3x^{2}-12}}$$

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

$$\frac{\frac{3\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}}-\frac{2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{x-10}{3x^{2}-12}}$$

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

$$\frac{\frac{3\left(x-2\right)-2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{x-10}{3x^{2}-12}}$$

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

$$\frac{\frac{3x-6-2x-4}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{x-10}{3x^{2}-12}}$$

Combine like terms in 3x-6-2x-4.

$$\frac{\frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{x-10}{3x^{2}-12}}$$

Divide \frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}} by \frac{x-10}{3x^{2}-12} by multiplying \frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}} by the reciprocal of \frac{x-10}{3x^{2}-12}.

$$\frac{\left(x-10\right)\left(3x^{2}-12\right)}{\left(x+2\right)\left(x-2\right)^{2}\left(x-10\right)}$$

Cancel out x-10 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

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

$$\frac{3}{x-2}$$