Solve 3x/x-2-4x/2-x-2x-1/x^2-4

$$\frac { 3 x } { x - 2 } - \frac { 4 x } { 2 - x } - \frac { 2 x - 1 } { x ^ { 2 } - 4 }$$

Evaluate

$$-\frac{7x^{2}+12x+1}{4-x^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and 2-x is x-2. Multiply \frac{4x}{2-x} times \frac{-1}{-1}.

$$\frac{3x}{x-2}-\frac{-4x}{x-2}-\frac{2x-1}{x^{2}-4}$$

Since \frac{3x}{x-2} and \frac{-4x}{x-2} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{3x+4x}{x-2}-\frac{2x-1}{x^{2}-4}$$

Combine like terms in 3x+4x.

$$\frac{7x}{x-2}-\frac{2x-1}{x^{2}-4}$$

Factor x^{2}-4.

$$\frac{7x}{x-2}-\frac{2x-1}{\left(x-2\right)\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and \left(x-2\right)\left(x+2\right) is \left(x-2\right)\left(x+2\right). Multiply \frac{7x}{x-2} times \frac{x+2}{x+2}.

$$\frac{7x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{2x-1}{\left(x-2\right)\left(x+2\right)}$$

Since \frac{7x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)} and \frac{2x-1}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7x\left(x+2\right)-\left(2x-1\right)}{\left(x-2\right)\left(x+2\right)}$$

Do the multiplications in 7x\left(x+2\right)-\left(2x-1\right).

$$\frac{7x^{2}+14x-2x+1}{\left(x-2\right)\left(x+2\right)}$$

Combine like terms in 7x^{2}+14x-2x+1.

$$\frac{7x^{2}+12x+1}{\left(x-2\right)\left(x+2\right)}$$

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

$$\frac{7x^{2}+12x+1}{x^{2}-4}$$

Expand

$$\frac{7x^{2}+12x+1}{x^{2}-4}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and 2-x is x-2. Multiply \frac{4x}{2-x} times \frac{-1}{-1}.

$$\frac{3x}{x-2}-\frac{-4x}{x-2}-\frac{2x-1}{x^{2}-4}$$

Since \frac{3x}{x-2} and \frac{-4x}{x-2} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{3x+4x}{x-2}-\frac{2x-1}{x^{2}-4}$$

Combine like terms in 3x+4x.

$$\frac{7x}{x-2}-\frac{2x-1}{x^{2}-4}$$

Factor x^{2}-4.

$$\frac{7x}{x-2}-\frac{2x-1}{\left(x-2\right)\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and \left(x-2\right)\left(x+2\right) is \left(x-2\right)\left(x+2\right). Multiply \frac{7x}{x-2} times \frac{x+2}{x+2}.

$$\frac{7x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{2x-1}{\left(x-2\right)\left(x+2\right)}$$

Since \frac{7x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)} and \frac{2x-1}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7x\left(x+2\right)-\left(2x-1\right)}{\left(x-2\right)\left(x+2\right)}$$

Do the multiplications in 7x\left(x+2\right)-\left(2x-1\right).

$$\frac{7x^{2}+14x-2x+1}{\left(x-2\right)\left(x+2\right)}$$

Combine like terms in 7x^{2}+14x-2x+1.

$$\frac{7x^{2}+12x+1}{\left(x-2\right)\left(x+2\right)}$$

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

$$\frac{7x^{2}+12x+1}{x^{2}-4}$$