Solve 2-x/x^2-7x+3/x+8-2x-4/x^2-36

$$\frac { 2 - x } { x ^ { 2 } - 7 x } + \frac { 3 } { x + 8 } - \frac { 2 x - 4 } { x ^ { 2 } - 36 }$$

Evaluate

$$\frac{-25x^{3}+60x^{2}+748x-576}{x\left(x-7\right)\left(x+8\right)\left(x^{2}-36\right)}$$

Short Solution Steps

Factor x^{2}-7x.

$$\frac{2-x}{x\left(x-7\right)}+\frac{3}{x+8}-\frac{2x-4}{x^{2}-36}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-7\right) and x+8 is x\left(x-7\right)\left(x+8\right). Multiply \frac{2-x}{x\left(x-7\right)} times \frac{x+8}{x+8}. Multiply \frac{3}{x+8} times \frac{x\left(x-7\right)}{x\left(x-7\right)}.

$$\frac{\left(2-x\right)\left(x+8\right)}{x\left(x-7\right)\left(x+8\right)}+\frac{3x\left(x-7\right)}{x\left(x-7\right)\left(x+8\right)}-\frac{2x-4}{x^{2}-36}$$

Since \frac{\left(2-x\right)\left(x+8\right)}{x\left(x-7\right)\left(x+8\right)} and \frac{3x\left(x-7\right)}{x\left(x-7\right)\left(x+8\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2-x\right)\left(x+8\right)+3x\left(x-7\right)}{x\left(x-7\right)\left(x+8\right)}-\frac{2x-4}{x^{2}-36}$$

Do the multiplications in \left(2-x\right)\left(x+8\right)+3x\left(x-7\right).

$$\frac{2x+16-x^{2}-8x+3x^{2}-21x}{x\left(x-7\right)\left(x+8\right)}-\frac{2x-4}{x^{2}-36}$$

Combine like terms in 2x+16-x^{2}-8x+3x^{2}-21x.

$$\frac{-27x+16+2x^{2}}{x\left(x-7\right)\left(x+8\right)}-\frac{2x-4}{x^{2}-36}$$

Factor x^{2}-36.

$$\frac{-27x+16+2x^{2}}{x\left(x-7\right)\left(x+8\right)}-\frac{2x-4}{\left(x-6\right)\left(x+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-7\right)\left(x+8\right) and \left(x-6\right)\left(x+6\right) is x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right). Multiply \frac{-27x+16+2x^{2}}{x\left(x-7\right)\left(x+8\right)} times \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}. Multiply \frac{2x-4}{\left(x-6\right)\left(x+6\right)} times \frac{x\left(x-7\right)\left(x+8\right)}{x\left(x-7\right)\left(x+8\right)}.

$$\frac{\left(-27x+16+2x^{2}\right)\left(x-6\right)\left(x+6\right)}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)}-\frac{\left(2x-4\right)x\left(x-7\right)\left(x+8\right)}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)}$$

Since \frac{\left(-27x+16+2x^{2}\right)\left(x-6\right)\left(x+6\right)}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)} and \frac{\left(2x-4\right)x\left(x-7\right)\left(x+8\right)}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-27x+16+2x^{2}\right)\left(x-6\right)\left(x+6\right)-\left(2x-4\right)x\left(x-7\right)\left(x+8\right)}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)}$$

Do the multiplications in \left(-27x+16+2x^{2}\right)\left(x-6\right)\left(x+6\right)-\left(2x-4\right)x\left(x-7\right)\left(x+8\right).

$$\frac{-27x^{3}+972x+16x^{2}-576+2x^{4}-72x^{2}-2x^{4}-2x^{3}+112x^{2}+4x^{3}+4x^{2}-224x}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)}$$

Combine like terms in -27x^{3}+972x+16x^{2}-576+2x^{4}-72x^{2}-2x^{4}-2x^{3}+112x^{2}+4x^{3}+4x^{2}-224x.

$$\frac{-25x^{3}+748x+60x^{2}-576}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)}$$

Expand x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right).

$$\frac{-25x^{3}+748x+60x^{2}-576}{x^{5}+x^{4}-92x^{3}-36x^{2}+2016x}$$

Expand

$$\frac{-25x^{3}+60x^{2}+748x-576}{\left(x+8\right)\left(x^{2}-36\right)\left(x^{2}-7x\right)}$$

Short Solution Steps

Factor x^{2}-7x.

$$\frac{2-x}{x\left(x-7\right)}+\frac{3}{x+8}-\frac{2x-4}{x^{2}-36}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-7\right) and x+8 is x\left(x-7\right)\left(x+8\right). Multiply \frac{2-x}{x\left(x-7\right)} times \frac{x+8}{x+8}. Multiply \frac{3}{x+8} times \frac{x\left(x-7\right)}{x\left(x-7\right)}.

$$\frac{\left(2-x\right)\left(x+8\right)}{x\left(x-7\right)\left(x+8\right)}+\frac{3x\left(x-7\right)}{x\left(x-7\right)\left(x+8\right)}-\frac{2x-4}{x^{2}-36}$$

Since \frac{\left(2-x\right)\left(x+8\right)}{x\left(x-7\right)\left(x+8\right)} and \frac{3x\left(x-7\right)}{x\left(x-7\right)\left(x+8\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2-x\right)\left(x+8\right)+3x\left(x-7\right)}{x\left(x-7\right)\left(x+8\right)}-\frac{2x-4}{x^{2}-36}$$

Do the multiplications in \left(2-x\right)\left(x+8\right)+3x\left(x-7\right).

$$\frac{2x+16-x^{2}-8x+3x^{2}-21x}{x\left(x-7\right)\left(x+8\right)}-\frac{2x-4}{x^{2}-36}$$

Combine like terms in 2x+16-x^{2}-8x+3x^{2}-21x.

$$\frac{-27x+16+2x^{2}}{x\left(x-7\right)\left(x+8\right)}-\frac{2x-4}{x^{2}-36}$$

Factor x^{2}-36.

$$\frac{-27x+16+2x^{2}}{x\left(x-7\right)\left(x+8\right)}-\frac{2x-4}{\left(x-6\right)\left(x+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-7\right)\left(x+8\right) and \left(x-6\right)\left(x+6\right) is x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right). Multiply \frac{-27x+16+2x^{2}}{x\left(x-7\right)\left(x+8\right)} times \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}. Multiply \frac{2x-4}{\left(x-6\right)\left(x+6\right)} times \frac{x\left(x-7\right)\left(x+8\right)}{x\left(x-7\right)\left(x+8\right)}.

$$\frac{\left(-27x+16+2x^{2}\right)\left(x-6\right)\left(x+6\right)}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)}-\frac{\left(2x-4\right)x\left(x-7\right)\left(x+8\right)}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)}$$

Since \frac{\left(-27x+16+2x^{2}\right)\left(x-6\right)\left(x+6\right)}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)} and \frac{\left(2x-4\right)x\left(x-7\right)\left(x+8\right)}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-27x+16+2x^{2}\right)\left(x-6\right)\left(x+6\right)-\left(2x-4\right)x\left(x-7\right)\left(x+8\right)}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)}$$

Do the multiplications in \left(-27x+16+2x^{2}\right)\left(x-6\right)\left(x+6\right)-\left(2x-4\right)x\left(x-7\right)\left(x+8\right).

$$\frac{-27x^{3}+972x+16x^{2}-576+2x^{4}-72x^{2}-2x^{4}-2x^{3}+112x^{2}+4x^{3}+4x^{2}-224x}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)}$$

Combine like terms in -27x^{3}+972x+16x^{2}-576+2x^{4}-72x^{2}-2x^{4}-2x^{3}+112x^{2}+4x^{3}+4x^{2}-224x.

$$\frac{-25x^{3}+748x+60x^{2}-576}{x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right)}$$

Expand x\left(x-7\right)\left(x-6\right)\left(x+6\right)\left(x+8\right).

$$\frac{-25x^{3}+748x+60x^{2}-576}{x^{5}+x^{4}-92x^{3}-36x^{2}+2016x}$$