Solve 1-x/x^2+x-6+2x-5/x^2-x-12-7x-10/x^2-6x+8

$$\frac { 1 - x } { x ^ { 2 } + x - 6 } + \frac { 2 x - 5 } { x ^ { 2 } - x - 12 } - \frac { 7 x - 10 } { x ^ { 2 } - 6 x + 8 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}+x-6. Factor x^{2}-x-12.

$$\frac{1-x}{\left(x-2\right)\left(x+3\right)}+\frac{2x-5}{\left(x-4\right)\left(x+3\right)}-\frac{7x-10}{x^{2}-6x+8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+3\right) and \left(x-4\right)\left(x+3\right) is \left(x-4\right)\left(x-2\right)\left(x+3\right). Multiply \frac{1-x}{\left(x-2\right)\left(x+3\right)} times \frac{x-4}{x-4}. Multiply \frac{2x-5}{\left(x-4\right)\left(x+3\right)} times \frac{x-2}{x-2}.

$$\frac{\left(1-x\right)\left(x-4\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}+\frac{\left(2x-5\right)\left(x-2\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{7x-10}{x^{2}-6x+8}$$

Since \frac{\left(1-x\right)\left(x-4\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)} and \frac{\left(2x-5\right)\left(x-2\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(1-x\right)\left(x-4\right)+\left(2x-5\right)\left(x-2\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{7x-10}{x^{2}-6x+8}$$

Do the multiplications in \left(1-x\right)\left(x-4\right)+\left(2x-5\right)\left(x-2\right).

$$\frac{x-4-x^{2}+4x+2x^{2}-4x-5x+10}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{7x-10}{x^{2}-6x+8}$$

Combine like terms in x-4-x^{2}+4x+2x^{2}-4x-5x+10.

$$\frac{-4x+6+x^{2}}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{7x-10}{x^{2}-6x+8}$$

Factor x^{2}-6x+8.

$$\frac{-4x+6+x^{2}}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{7x-10}{\left(x-4\right)\left(x-2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4\right)\left(x-2\right)\left(x+3\right) and \left(x-4\right)\left(x-2\right) is \left(x-4\right)\left(x-2\right)\left(x+3\right). Multiply \frac{7x-10}{\left(x-4\right)\left(x-2\right)} times \frac{x+3}{x+3}.

$$\frac{-4x+6+x^{2}}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{\left(7x-10\right)\left(x+3\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}$$

Since \frac{-4x+6+x^{2}}{\left(x-4\right)\left(x-2\right)\left(x+3\right)} and \frac{\left(7x-10\right)\left(x+3\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x+6+x^{2}-\left(7x-10\right)\left(x+3\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}$$

Do the multiplications in -4x+6+x^{2}-\left(7x-10\right)\left(x+3\right).

$$\frac{-4x+6+x^{2}-7x^{2}-21x+10x+30}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}$$

Combine like terms in -4x+6+x^{2}-7x^{2}-21x+10x+30.

$$\frac{-15x+36-6x^{2}}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}$$

Expand \left(x-4\right)\left(x-2\right)\left(x+3\right).

$$\frac{-15x+36-6x^{2}}{x^{3}-3x^{2}-10x+24}$$

Expand

$$-\frac{3\left(2x^{2}+5x-12\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}+x-6. Factor x^{2}-x-12.

$$\frac{1-x}{\left(x-2\right)\left(x+3\right)}+\frac{2x-5}{\left(x-4\right)\left(x+3\right)}-\frac{7x-10}{x^{2}-6x+8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+3\right) and \left(x-4\right)\left(x+3\right) is \left(x-4\right)\left(x-2\right)\left(x+3\right). Multiply \frac{1-x}{\left(x-2\right)\left(x+3\right)} times \frac{x-4}{x-4}. Multiply \frac{2x-5}{\left(x-4\right)\left(x+3\right)} times \frac{x-2}{x-2}.

$$\frac{\left(1-x\right)\left(x-4\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}+\frac{\left(2x-5\right)\left(x-2\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{7x-10}{x^{2}-6x+8}$$

Since \frac{\left(1-x\right)\left(x-4\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)} and \frac{\left(2x-5\right)\left(x-2\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(1-x\right)\left(x-4\right)+\left(2x-5\right)\left(x-2\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{7x-10}{x^{2}-6x+8}$$

Do the multiplications in \left(1-x\right)\left(x-4\right)+\left(2x-5\right)\left(x-2\right).

$$\frac{x-4-x^{2}+4x+2x^{2}-4x-5x+10}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{7x-10}{x^{2}-6x+8}$$

Combine like terms in x-4-x^{2}+4x+2x^{2}-4x-5x+10.

$$\frac{-4x+6+x^{2}}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{7x-10}{x^{2}-6x+8}$$

Factor x^{2}-6x+8.

$$\frac{-4x+6+x^{2}}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{7x-10}{\left(x-4\right)\left(x-2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4\right)\left(x-2\right)\left(x+3\right) and \left(x-4\right)\left(x-2\right) is \left(x-4\right)\left(x-2\right)\left(x+3\right). Multiply \frac{7x-10}{\left(x-4\right)\left(x-2\right)} times \frac{x+3}{x+3}.

$$\frac{-4x+6+x^{2}}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}-\frac{\left(7x-10\right)\left(x+3\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}$$

Since \frac{-4x+6+x^{2}}{\left(x-4\right)\left(x-2\right)\left(x+3\right)} and \frac{\left(7x-10\right)\left(x+3\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x+6+x^{2}-\left(7x-10\right)\left(x+3\right)}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}$$

Do the multiplications in -4x+6+x^{2}-\left(7x-10\right)\left(x+3\right).

$$\frac{-4x+6+x^{2}-7x^{2}-21x+10x+30}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}$$

Combine like terms in -4x+6+x^{2}-7x^{2}-21x+10x+30.

$$\frac{-15x+36-6x^{2}}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}$$

Expand \left(x-4\right)\left(x-2\right)\left(x+3\right).

$$\frac{-15x+36-6x^{2}}{x^{3}-3x^{2}-10x+24}$$