Solve 3x^2+4x/x^2-2x-2x-7/x-x+8/x-2

$$\frac { 3 x ^ { 2 } + 4 x } { x ^ { 2 } - 2 x } - \frac { 2 x - 7 } { x } - \frac { x + 8 } { x - 2 }$$

Evaluate

$$\frac{7}{x}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{3x^{2}+4x}{x^{2}-2x}.

$$\frac{x\left(3x+4\right)}{x\left(x-2\right)}-\frac{2x-7}{x}-\frac{x+8}{x-2}$$

Cancel out x in both numerator and denominator.

$$\frac{3x+4}{x-2}-\frac{2x-7}{x}-\frac{x+8}{x-2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x is x\left(x-2\right). Multiply \frac{3x+4}{x-2} times \frac{x}{x}. Multiply \frac{2x-7}{x} times \frac{x-2}{x-2}.

$$\frac{\left(3x+4\right)x}{x\left(x-2\right)}-\frac{\left(2x-7\right)\left(x-2\right)}{x\left(x-2\right)}-\frac{x+8}{x-2}$$

Since \frac{\left(3x+4\right)x}{x\left(x-2\right)} and \frac{\left(2x-7\right)\left(x-2\right)}{x\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3x+4\right)x-\left(2x-7\right)\left(x-2\right)}{x\left(x-2\right)}-\frac{x+8}{x-2}$$

Do the multiplications in \left(3x+4\right)x-\left(2x-7\right)\left(x-2\right).

$$\frac{3x^{2}+4x-2x^{2}+4x+7x-14}{x\left(x-2\right)}-\frac{x+8}{x-2}$$

Combine like terms in 3x^{2}+4x-2x^{2}+4x+7x-14.

$$\frac{x^{2}+15x-14}{x\left(x-2\right)}-\frac{x+8}{x-2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-2\right) and x-2 is x\left(x-2\right). Multiply \frac{x+8}{x-2} times \frac{x}{x}.

$$\frac{x^{2}+15x-14}{x\left(x-2\right)}-\frac{\left(x+8\right)x}{x\left(x-2\right)}$$

Since \frac{x^{2}+15x-14}{x\left(x-2\right)} and \frac{\left(x+8\right)x}{x\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+15x-14-\left(x+8\right)x}{x\left(x-2\right)}$$

Do the multiplications in x^{2}+15x-14-\left(x+8\right)x.

$$\frac{x^{2}+15x-14-x^{2}-8x}{x\left(x-2\right)}$$

Combine like terms in x^{2}+15x-14-x^{2}-8x.

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

Factor the expressions that are not already factored in \frac{7x-14}{x\left(x-2\right)}.

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

Cancel out x-2 in both numerator and denominator.

$$\frac{7}{x}$$

Expand

$$\frac{7}{x}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{3x^{2}+4x}{x^{2}-2x}.

$$\frac{x\left(3x+4\right)}{x\left(x-2\right)}-\frac{2x-7}{x}-\frac{x+8}{x-2}$$

Cancel out x in both numerator and denominator.

$$\frac{3x+4}{x-2}-\frac{2x-7}{x}-\frac{x+8}{x-2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x is x\left(x-2\right). Multiply \frac{3x+4}{x-2} times \frac{x}{x}. Multiply \frac{2x-7}{x} times \frac{x-2}{x-2}.

$$\frac{\left(3x+4\right)x}{x\left(x-2\right)}-\frac{\left(2x-7\right)\left(x-2\right)}{x\left(x-2\right)}-\frac{x+8}{x-2}$$

Since \frac{\left(3x+4\right)x}{x\left(x-2\right)} and \frac{\left(2x-7\right)\left(x-2\right)}{x\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3x+4\right)x-\left(2x-7\right)\left(x-2\right)}{x\left(x-2\right)}-\frac{x+8}{x-2}$$

Do the multiplications in \left(3x+4\right)x-\left(2x-7\right)\left(x-2\right).

$$\frac{3x^{2}+4x-2x^{2}+4x+7x-14}{x\left(x-2\right)}-\frac{x+8}{x-2}$$

Combine like terms in 3x^{2}+4x-2x^{2}+4x+7x-14.

$$\frac{x^{2}+15x-14}{x\left(x-2\right)}-\frac{x+8}{x-2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-2\right) and x-2 is x\left(x-2\right). Multiply \frac{x+8}{x-2} times \frac{x}{x}.

$$\frac{x^{2}+15x-14}{x\left(x-2\right)}-\frac{\left(x+8\right)x}{x\left(x-2\right)}$$

Since \frac{x^{2}+15x-14}{x\left(x-2\right)} and \frac{\left(x+8\right)x}{x\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+15x-14-\left(x+8\right)x}{x\left(x-2\right)}$$

Do the multiplications in x^{2}+15x-14-\left(x+8\right)x.

$$\frac{x^{2}+15x-14-x^{2}-8x}{x\left(x-2\right)}$$

Combine like terms in x^{2}+15x-14-x^{2}-8x.

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

Factor the expressions that are not already factored in \frac{7x-14}{x\left(x-2\right)}.

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

Cancel out x-2 in both numerator and denominator.

$$\frac{7}{x}$$