Solve x^2+4x/x^2-4:(x/x-2+x/x+2-24/3x^2-12)

$$\frac { x ^ { 2 } + 4 x } { x ^ { 2 } - 4 } : ( \frac { x } { x - 2 } + \frac { x } { x + 2 } - \frac { 24 } { 3 x ^ { 2 } - 12 } )$$

Evaluate

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

Short Solution Steps

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

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\frac{x\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{24}{3x^{2}-12}}$$

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

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{x\left(x+2\right)+x\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{24}{3x^{2}-12}}$$

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

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{x^{2}+2x+x^{2}-2x}{\left(x-2\right)\left(x+2\right)}-\frac{24}{3x^{2}-12}}$$

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

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{2x^{2}}{\left(x-2\right)\left(x+2\right)}-\frac{24}{3x^{2}-12}}$$

Factor 3x^{2}-12.

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{2x^{2}}{\left(x-2\right)\left(x+2\right)}-\frac{24}{3\left(x-2\right)\left(x+2\right)}}$$

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

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{3\times 2x^{2}}{3\left(x-2\right)\left(x+2\right)}-\frac{24}{3\left(x-2\right)\left(x+2\right)}}$$

Since \frac{3\times 2x^{2}}{3\left(x-2\right)\left(x+2\right)} and \frac{24}{3\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{3\times 2x^{2}-24}{3\left(x-2\right)\left(x+2\right)}}$$

Do the multiplications in 3\times 2x^{2}-24.

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{6x^{2}-24}{3\left(x-2\right)\left(x+2\right)}}$$

Factor the expressions that are not already factored in \frac{6x^{2}-24}{3\left(x-2\right)\left(x+2\right)}.

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

Cancel out 3\left(x-2\right)\left(x+2\right) in both numerator and denominator.

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

Express \frac{\frac{x^{2}+4x}{x^{2}-4}}{2} as a single fraction.

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

Use the distributive property to multiply x^{2}-4 by 2.

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

Expand

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

Short Solution Steps

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

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\frac{x\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{24}{3x^{2}-12}}$$

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

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{x\left(x+2\right)+x\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{24}{3x^{2}-12}}$$

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

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{x^{2}+2x+x^{2}-2x}{\left(x-2\right)\left(x+2\right)}-\frac{24}{3x^{2}-12}}$$

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

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{2x^{2}}{\left(x-2\right)\left(x+2\right)}-\frac{24}{3x^{2}-12}}$$

Factor 3x^{2}-12.

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{2x^{2}}{\left(x-2\right)\left(x+2\right)}-\frac{24}{3\left(x-2\right)\left(x+2\right)}}$$

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

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{3\times 2x^{2}}{3\left(x-2\right)\left(x+2\right)}-\frac{24}{3\left(x-2\right)\left(x+2\right)}}$$

Since \frac{3\times 2x^{2}}{3\left(x-2\right)\left(x+2\right)} and \frac{24}{3\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{3\times 2x^{2}-24}{3\left(x-2\right)\left(x+2\right)}}$$

Do the multiplications in 3\times 2x^{2}-24.

$$\frac{\frac{x^{2}+4x}{x^{2}-4}}{\frac{6x^{2}-24}{3\left(x-2\right)\left(x+2\right)}}$$

Factor the expressions that are not already factored in \frac{6x^{2}-24}{3\left(x-2\right)\left(x+2\right)}.

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

Cancel out 3\left(x-2\right)\left(x+2\right) in both numerator and denominator.

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

Express \frac{\frac{x^{2}+4x}{x^{2}-4}}{2} as a single fraction.

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

Use the distributive property to multiply x^{2}-4 by 2.

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