Solve x-2/frac{x-2{2x-6}div(-5/x-3-x-3)}

$$\frac { x - 2 } { \frac { x - 2 } { 2 x - 6 } \div ( \frac { - 5 } { x - 3 } - x - 3 ) }$$

Evaluate

$$8-2x^{2}$$

Short Solution Steps

Divide x-2 by \frac{\frac{x-2}{2x-6}}{\frac{-5}{x-3}-x-3} by multiplying x-2 by the reciprocal of \frac{\frac{x-2}{2x-6}}{\frac{-5}{x-3}-x-3}.

$$\frac{\left(x-2\right)\left(\frac{-5}{x-3}-x-3\right)}{\frac{x-2}{2x-6}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -x-3 times \frac{x-3}{x-3}.

$$\frac{\left(x-2\right)\left(\frac{-5}{x-3}+\frac{\left(-x-3\right)\left(x-3\right)}{x-3}\right)}{\frac{x-2}{2x-6}}$$

Since \frac{-5}{x-3} and \frac{\left(-x-3\right)\left(x-3\right)}{x-3} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-2\right)\times \frac{-5+\left(-x-3\right)\left(x-3\right)}{x-3}}{\frac{x-2}{2x-6}}$$

Do the multiplications in -5+\left(-x-3\right)\left(x-3\right).

$$\frac{\left(x-2\right)\times \frac{-5-x^{2}+3x-3x+9}{x-3}}{\frac{x-2}{2x-6}}$$

Combine like terms in -5-x^{2}+3x-3x+9.

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

Express \left(x-2\right)\times \frac{4-x^{2}}{x-3} as a single fraction.

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

Divide \frac{\left(x-2\right)\left(4-x^{2}\right)}{x-3} by \frac{x-2}{2x-6} by multiplying \frac{\left(x-2\right)\left(4-x^{2}\right)}{x-3} by the reciprocal of \frac{x-2}{2x-6}.

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

Cancel out x-2 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-3 in both numerator and denominator.

$$2\left(x-2\right)\left(-x-2\right)$$

Expand the expression.

$$-2x^{2}+8$$

Expand

$$8-2x^{2}$$

Short Solution Steps

Divide x-2 by \frac{\frac{x-2}{2x-6}}{\frac{-5}{x-3}-x-3} by multiplying x-2 by the reciprocal of \frac{\frac{x-2}{2x-6}}{\frac{-5}{x-3}-x-3}.

$$\frac{\left(x-2\right)\left(\frac{-5}{x-3}-x-3\right)}{\frac{x-2}{2x-6}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -x-3 times \frac{x-3}{x-3}.

$$\frac{\left(x-2\right)\left(\frac{-5}{x-3}+\frac{\left(-x-3\right)\left(x-3\right)}{x-3}\right)}{\frac{x-2}{2x-6}}$$

Since \frac{-5}{x-3} and \frac{\left(-x-3\right)\left(x-3\right)}{x-3} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-2\right)\times \frac{-5+\left(-x-3\right)\left(x-3\right)}{x-3}}{\frac{x-2}{2x-6}}$$

Do the multiplications in -5+\left(-x-3\right)\left(x-3\right).

$$\frac{\left(x-2\right)\times \frac{-5-x^{2}+3x-3x+9}{x-3}}{\frac{x-2}{2x-6}}$$

Combine like terms in -5-x^{2}+3x-3x+9.

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

Express \left(x-2\right)\times \frac{4-x^{2}}{x-3} as a single fraction.

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

Divide \frac{\left(x-2\right)\left(4-x^{2}\right)}{x-3} by \frac{x-2}{2x-6} by multiplying \frac{\left(x-2\right)\left(4-x^{2}\right)}{x-3} by the reciprocal of \frac{x-2}{2x-6}.

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

Cancel out x-2 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-3 in both numerator and denominator.

$$2\left(x-2\right)\left(-x-2\right)$$

Expand the expression.

$$-2x^{2}+8$$