Solve x/x^2-8x+7-x/x^2-4x-21

$$\frac { x } { x ^ { 2 } - 8 x + 7 } - \frac { x } { x ^ { 2 } - 4 x - 21 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-8x+7. Factor x^{2}-4x-21.

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

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

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

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

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

Do the multiplications in x\left(x+3\right)-x\left(x-1\right).

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

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

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

Expand \left(x-7\right)\left(x-1\right)\left(x+3\right).

$$\frac{4x}{x^{3}-5x^{2}-17x+21}$$

Factor

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