Solve 3/2-3/2-x-2/x^2-4

$$\frac { 3 } { 2 } - \frac { 3 } { 2 - x } - \frac { 2 } { x ^ { 2 } - 4 }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in -3x+6-6.

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

Factor x^{2}-4.

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

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

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

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

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

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

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

Expand 2\left(x-2\right)\left(x+2\right).

$$\frac{3x^{2}+6x-4}{2x^{2}-8}$$

Factor

$$-\frac{3\left(x-\left(-\frac{\sqrt{21}}{3}-1\right)\right)\left(x-\left(\frac{\sqrt{21}}{3}-1\right)\right)}{2\left(4-x^{2}\right)}$$