Solve 3/2x-8+3-2x/2x^2+8x-15x-12/2

$$\frac { 3 } { 2 x - 8 } + \frac { 3 - 2 x } { 2 x ^ { 2 } + 8 x } - \frac { 15 x - 12 } { 2 }$$

Evaluate

$$-\frac{15x^{4}-12x^{3}-241x^{2}+169x+12}{2x\left(x^{2}-16\right)}$$

Short Solution Steps

Factor 2x-8. Factor 2x^{2}+8x.

$$\frac{3}{2\left(x-4\right)}+\frac{3-2x}{2x\left(x+4\right)}-\frac{15x-12}{2}$$

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

$$\frac{3x\left(x+4\right)}{2x\left(x-4\right)\left(x+4\right)}+\frac{\left(3-2x\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)}-\frac{15x-12}{2}$$

Since \frac{3x\left(x+4\right)}{2x\left(x-4\right)\left(x+4\right)} and \frac{\left(3-2x\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{3x\left(x+4\right)+\left(3-2x\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)}-\frac{15x-12}{2}$$

Do the multiplications in 3x\left(x+4\right)+\left(3-2x\right)\left(x-4\right).

$$\frac{3x^{2}+12x+3x-12-2x^{2}+8x}{2x\left(x-4\right)\left(x+4\right)}-\frac{15x-12}{2}$$

Combine like terms in 3x^{2}+12x+3x-12-2x^{2}+8x.

$$\frac{x^{2}+23x-12}{2x\left(x-4\right)\left(x+4\right)}-\frac{15x-12}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x\left(x-4\right)\left(x+4\right) and 2 is 2x\left(x-4\right)\left(x+4\right). Multiply \frac{15x-12}{2} times \frac{x\left(x-4\right)\left(x+4\right)}{x\left(x-4\right)\left(x+4\right)}.

$$\frac{x^{2}+23x-12}{2x\left(x-4\right)\left(x+4\right)}-\frac{\left(15x-12\right)x\left(x-4\right)\left(x+4\right)}{2x\left(x-4\right)\left(x+4\right)}$$

Since \frac{x^{2}+23x-12}{2x\left(x-4\right)\left(x+4\right)} and \frac{\left(15x-12\right)x\left(x-4\right)\left(x+4\right)}{2x\left(x-4\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+23x-12-\left(15x-12\right)x\left(x-4\right)\left(x+4\right)}{2x\left(x-4\right)\left(x+4\right)}$$

Do the multiplications in x^{2}+23x-12-\left(15x-12\right)x\left(x-4\right)\left(x+4\right).

$$\frac{x^{2}+23x-12-15x^{4}+240x^{2}+12x^{3}-192x}{2x\left(x-4\right)\left(x+4\right)}$$

Combine like terms in x^{2}+23x-12-15x^{4}+240x^{2}+12x^{3}-192x.

$$\frac{241x^{2}-169x-12-15x^{4}+12x^{3}}{2x\left(x-4\right)\left(x+4\right)}$$

Expand 2x\left(x-4\right)\left(x+4\right).

$$\frac{241x^{2}-169x-12-15x^{4}+12x^{3}}{2x^{3}-32x}$$

Expand

$$-\frac{15x^{4}-12x^{3}-241x^{2}+169x+12}{2\left(x-4\right)\left(x^{2}+4x\right)}$$

Short Solution Steps

Factor 2x-8. Factor 2x^{2}+8x.

$$\frac{3}{2\left(x-4\right)}+\frac{3-2x}{2x\left(x+4\right)}-\frac{15x-12}{2}$$

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

$$\frac{3x\left(x+4\right)}{2x\left(x-4\right)\left(x+4\right)}+\frac{\left(3-2x\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)}-\frac{15x-12}{2}$$

Since \frac{3x\left(x+4\right)}{2x\left(x-4\right)\left(x+4\right)} and \frac{\left(3-2x\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{3x\left(x+4\right)+\left(3-2x\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)}-\frac{15x-12}{2}$$

Do the multiplications in 3x\left(x+4\right)+\left(3-2x\right)\left(x-4\right).

$$\frac{3x^{2}+12x+3x-12-2x^{2}+8x}{2x\left(x-4\right)\left(x+4\right)}-\frac{15x-12}{2}$$

Combine like terms in 3x^{2}+12x+3x-12-2x^{2}+8x.

$$\frac{x^{2}+23x-12}{2x\left(x-4\right)\left(x+4\right)}-\frac{15x-12}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x\left(x-4\right)\left(x+4\right) and 2 is 2x\left(x-4\right)\left(x+4\right). Multiply \frac{15x-12}{2} times \frac{x\left(x-4\right)\left(x+4\right)}{x\left(x-4\right)\left(x+4\right)}.

$$\frac{x^{2}+23x-12}{2x\left(x-4\right)\left(x+4\right)}-\frac{\left(15x-12\right)x\left(x-4\right)\left(x+4\right)}{2x\left(x-4\right)\left(x+4\right)}$$

Since \frac{x^{2}+23x-12}{2x\left(x-4\right)\left(x+4\right)} and \frac{\left(15x-12\right)x\left(x-4\right)\left(x+4\right)}{2x\left(x-4\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+23x-12-\left(15x-12\right)x\left(x-4\right)\left(x+4\right)}{2x\left(x-4\right)\left(x+4\right)}$$

Do the multiplications in x^{2}+23x-12-\left(15x-12\right)x\left(x-4\right)\left(x+4\right).

$$\frac{x^{2}+23x-12-15x^{4}+240x^{2}+12x^{3}-192x}{2x\left(x-4\right)\left(x+4\right)}$$

Combine like terms in x^{2}+23x-12-15x^{4}+240x^{2}+12x^{3}-192x.

$$\frac{241x^{2}-169x-12-15x^{4}+12x^{3}}{2x\left(x-4\right)\left(x+4\right)}$$

Expand 2x\left(x-4\right)\left(x+4\right).

$$\frac{241x^{2}-169x-12-15x^{4}+12x^{3}}{2x^{3}-32x}$$