Solve 4x/x^2+6x-7-3x/x^2+x-2

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

Evaluate

$$\frac{x\left(x-13\right)}{\left(x-1\right)\left(x+2\right)\left(x+7\right)}$$

Short Solution Steps

Factor x^{2}+6x-7. Factor x^{2}+x-2.

$$\frac{4x}{\left(x-1\right)\left(x+7\right)}-\frac{3x}{\left(x-1\right)\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+7\right) and \left(x-1\right)\left(x+2\right) is \left(x-1\right)\left(x+2\right)\left(x+7\right). Multiply \frac{4x}{\left(x-1\right)\left(x+7\right)} times \frac{x+2}{x+2}. Multiply \frac{3x}{\left(x-1\right)\left(x+2\right)} times \frac{x+7}{x+7}.

$$\frac{4x\left(x+2\right)}{\left(x-1\right)\left(x+2\right)\left(x+7\right)}-\frac{3x\left(x+7\right)}{\left(x-1\right)\left(x+2\right)\left(x+7\right)}$$

Since \frac{4x\left(x+2\right)}{\left(x-1\right)\left(x+2\right)\left(x+7\right)} and \frac{3x\left(x+7\right)}{\left(x-1\right)\left(x+2\right)\left(x+7\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4x\left(x+2\right)-3x\left(x+7\right)}{\left(x-1\right)\left(x+2\right)\left(x+7\right)}$$

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

$$\frac{4x^{2}+8x-3x^{2}-21x}{\left(x-1\right)\left(x+2\right)\left(x+7\right)}$$

Combine like terms in 4x^{2}+8x-3x^{2}-21x.

$$\frac{x^{2}-13x}{\left(x-1\right)\left(x+2\right)\left(x+7\right)}$$

Expand \left(x-1\right)\left(x+2\right)\left(x+7\right).

$$\frac{x^{2}-13x}{x^{3}+8x^{2}+5x-14}$$

Factor

$$\frac{x\left(x-13\right)}{\left(x-1\right)\left(x+2\right)\left(x+7\right)}$$