Solve x/x^2-3x-10-2x/2x^2+x-6

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

Evaluate

$$\frac{7x}{\left(x-5\right)\left(2x-3\right)\left(x+2\right)}$$

Short Solution Steps

Factor x^{2}-3x-10. Factor 2x^{2}+x-6.

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

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

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

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

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

Do the multiplications in x\left(2x-3\right)-2x\left(x-5\right).

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

Combine like terms in 2x^{2}-3x-2x^{2}+10x.

$$\frac{7x}{\left(x-5\right)\left(2x-3\right)\left(x+2\right)}$$

Expand \left(x-5\right)\left(2x-3\right)\left(x+2\right).

$$\frac{7x}{2x^{3}-9x^{2}-11x+30}$$

Factor

$$\frac{7x}{\left(x-5\right)\left(2x-3\right)\left(x+2\right)}$$