Solve x+2+3/x-2/x-6+frac{7{x}}

$$\frac { x + 2 + \frac { 3 } { x - 2 } } { x - 6 + \frac { 7 } { x } }$$

Evaluate

$$\frac{x\left(x^{2}-1\right)}{\left(x-2\right)\left(x^{2}-6x+7\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x+2 times \frac{x-2}{x-2}.

$$\frac{\frac{\left(x+2\right)\left(x-2\right)}{x-2}+\frac{3}{x-2}}{x-6+\frac{7}{x}}$$

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

$$\frac{\frac{\left(x+2\right)\left(x-2\right)+3}{x-2}}{x-6+\frac{7}{x}}$$

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

$$\frac{\frac{x^{2}-2x+2x-4+3}{x-2}}{x-6+\frac{7}{x}}$$

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

$$\frac{\frac{x^{2}-1}{x-2}}{x-6+\frac{7}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x-6 times \frac{x}{x}.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{\left(x-6\right)x}{x}+\frac{7}{x}}$$

Since \frac{\left(x-6\right)x}{x} and \frac{7}{x} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{\left(x-6\right)x+7}{x}}$$

Do the multiplications in \left(x-6\right)x+7.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{x^{2}-6x+7}{x}}$$

Divide \frac{x^{2}-1}{x-2} by \frac{x^{2}-6x+7}{x} by multiplying \frac{x^{2}-1}{x-2} by the reciprocal of \frac{x^{2}-6x+7}{x}.

$$\frac{\left(x^{2}-1\right)x}{\left(x-2\right)\left(x^{2}-6x+7\right)}$$

Use the distributive property to multiply x^{2}-1 by x.

$$\frac{x^{3}-x}{\left(x-2\right)\left(x^{2}-6x+7\right)}$$

Apply the distributive property by multiplying each term of x-2 by each term of x^{2}-6x+7.

$$\frac{x^{3}-x}{x^{3}-6x^{2}+7x-2x^{2}+12x-14}$$

Combine -6x^{2} and -2x^{2} to get -8x^{2}.

$$\frac{x^{3}-x}{x^{3}-8x^{2}+7x+12x-14}$$

Combine 7x and 12x to get 19x.

$$\frac{x^{3}-x}{x^{3}-8x^{2}+19x-14}$$

Expand

$$\frac{x^{3}-x}{\left(x-2\right)\left(x^{2}-6x+7\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x+2 times \frac{x-2}{x-2}.

$$\frac{\frac{\left(x+2\right)\left(x-2\right)}{x-2}+\frac{3}{x-2}}{x-6+\frac{7}{x}}$$

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

$$\frac{\frac{\left(x+2\right)\left(x-2\right)+3}{x-2}}{x-6+\frac{7}{x}}$$

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

$$\frac{\frac{x^{2}-2x+2x-4+3}{x-2}}{x-6+\frac{7}{x}}$$

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

$$\frac{\frac{x^{2}-1}{x-2}}{x-6+\frac{7}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x-6 times \frac{x}{x}.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{\left(x-6\right)x}{x}+\frac{7}{x}}$$

Since \frac{\left(x-6\right)x}{x} and \frac{7}{x} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{\left(x-6\right)x+7}{x}}$$

Do the multiplications in \left(x-6\right)x+7.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{x^{2}-6x+7}{x}}$$

Divide \frac{x^{2}-1}{x-2} by \frac{x^{2}-6x+7}{x} by multiplying \frac{x^{2}-1}{x-2} by the reciprocal of \frac{x^{2}-6x+7}{x}.

$$\frac{\left(x^{2}-1\right)x}{\left(x-2\right)\left(x^{2}-6x+7\right)}$$

Use the distributive property to multiply x^{2}-1 by x.

$$\frac{x^{3}-x}{\left(x-2\right)\left(x^{2}-6x+7\right)}$$

Apply the distributive property by multiplying each term of x-2 by each term of x^{2}-6x+7.

$$\frac{x^{3}-x}{x^{3}-6x^{2}+7x-2x^{2}+12x-14}$$

Combine -6x^{2} and -2x^{2} to get -8x^{2}.

$$\frac{x^{3}-x}{x^{3}-8x^{2}+7x+12x-14}$$

Combine 7x and 12x to get 19x.

$$\frac{x^{3}-x}{x^{3}-8x^{2}+19x-14}$$