Solve x+11/2x^2-7x-30-x-3/2x^2+21x+40

$$\frac { x + 11 } { 2 x ^ { 2 } - 7 x - 30 } - \frac { x - 3 } { 2 x ^ { 2 } + 21 x + 40 }$$

Evaluate

$$\frac{14}{\left(x-6\right)\left(x+8\right)}$$

Short Solution Steps

Factor 2x^{2}-7x-30. Factor 2x^{2}+21x+40.

$$\frac{x+11}{\left(x-6\right)\left(2x+5\right)}-\frac{x-3}{\left(x+8\right)\left(2x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-6\right)\left(2x+5\right) and \left(x+8\right)\left(2x+5\right) is \left(x-6\right)\left(x+8\right)\left(2x+5\right). Multiply \frac{x+11}{\left(x-6\right)\left(2x+5\right)} times \frac{x+8}{x+8}. Multiply \frac{x-3}{\left(x+8\right)\left(2x+5\right)} times \frac{x-6}{x-6}.

$$\frac{\left(x+11\right)\left(x+8\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}-\frac{\left(x-3\right)\left(x-6\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}$$

Since \frac{\left(x+11\right)\left(x+8\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)} and \frac{\left(x-3\right)\left(x-6\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+11\right)\left(x+8\right)-\left(x-3\right)\left(x-6\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}$$

Do the multiplications in \left(x+11\right)\left(x+8\right)-\left(x-3\right)\left(x-6\right).

$$\frac{x^{2}+8x+11x+88-x^{2}+6x+3x-18}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}$$

Combine like terms in x^{2}+8x+11x+88-x^{2}+6x+3x-18.

$$\frac{28x+70}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}$$

Factor the expressions that are not already factored in \frac{28x+70}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}.

$$\frac{14\left(2x+5\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}$$

Cancel out 2x+5 in both numerator and denominator.

$$\frac{14}{\left(x-6\right)\left(x+8\right)}$$

Expand \left(x-6\right)\left(x+8\right).

$$\frac{14}{x^{2}+2x-48}$$

Expand

$$\frac{56}{\left(2x-12\right)\left(2x+16\right)}$$

Short Solution Steps

Factor 2x^{2}-7x-30. Factor 2x^{2}+21x+40.

$$\frac{x+11}{\left(x-6\right)\left(2x+5\right)}-\frac{x-3}{\left(x+8\right)\left(2x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-6\right)\left(2x+5\right) and \left(x+8\right)\left(2x+5\right) is \left(x-6\right)\left(x+8\right)\left(2x+5\right). Multiply \frac{x+11}{\left(x-6\right)\left(2x+5\right)} times \frac{x+8}{x+8}. Multiply \frac{x-3}{\left(x+8\right)\left(2x+5\right)} times \frac{x-6}{x-6}.

$$\frac{\left(x+11\right)\left(x+8\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}-\frac{\left(x-3\right)\left(x-6\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}$$

Since \frac{\left(x+11\right)\left(x+8\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)} and \frac{\left(x-3\right)\left(x-6\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+11\right)\left(x+8\right)-\left(x-3\right)\left(x-6\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}$$

Do the multiplications in \left(x+11\right)\left(x+8\right)-\left(x-3\right)\left(x-6\right).

$$\frac{x^{2}+8x+11x+88-x^{2}+6x+3x-18}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}$$

Combine like terms in x^{2}+8x+11x+88-x^{2}+6x+3x-18.

$$\frac{28x+70}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}$$

Factor the expressions that are not already factored in \frac{28x+70}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}.

$$\frac{14\left(2x+5\right)}{\left(x-6\right)\left(x+8\right)\left(2x+5\right)}$$

Cancel out 2x+5 in both numerator and denominator.

$$\frac{14}{\left(x-6\right)\left(x+8\right)}$$

Expand \left(x-6\right)\left(x+8\right).

$$\frac{14}{x^{2}+2x-48}$$