Solve x-3/4x^2+37x-30-x+2/x^2-100

$$\frac { x - 3 } { 4 x ^ { 2 } + 37 x - 30 } - \frac { x + 2 } { x ^ { 2 } - 100 }$$

Evaluate

$$-\frac{3\left(x^{2}+6x-12\right)}{\left(4x-3\right)\left(x^{2}-100\right)}$$

Short Solution Steps

Factor 4x^{2}+37x-30. Factor x^{2}-100.

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

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

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

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

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

Do the multiplications in \left(x-3\right)\left(x-10\right)-\left(x+2\right)\left(4x-3\right).

$$\frac{x^{2}-10x-3x+30-4x^{2}+3x-8x+6}{\left(x-10\right)\left(4x-3\right)\left(x+10\right)}$$

Combine like terms in x^{2}-10x-3x+30-4x^{2}+3x-8x+6.

$$\frac{-3x^{2}-18x+36}{\left(x-10\right)\left(4x-3\right)\left(x+10\right)}$$

Expand \left(x-10\right)\left(4x-3\right)\left(x+10\right).

$$\frac{-3x^{2}-18x+36}{4x^{3}-3x^{2}-400x+300}$$

Expand

$$-\frac{3\left(x^{2}+6x-12\right)}{\left(4x-3\right)\left(x^{2}-100\right)}$$

Short Solution Steps

Factor 4x^{2}+37x-30. Factor x^{2}-100.

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

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

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

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

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

Do the multiplications in \left(x-3\right)\left(x-10\right)-\left(x+2\right)\left(4x-3\right).

$$\frac{x^{2}-10x-3x+30-4x^{2}+3x-8x+6}{\left(x-10\right)\left(4x-3\right)\left(x+10\right)}$$

Combine like terms in x^{2}-10x-3x+30-4x^{2}+3x-8x+6.

$$\frac{-3x^{2}-18x+36}{\left(x-10\right)\left(4x-3\right)\left(x+10\right)}$$

Expand \left(x-10\right)\left(4x-3\right)\left(x+10\right).

$$\frac{-3x^{2}-18x+36}{4x^{3}-3x^{2}-400x+300}$$