Solve 1/2x^2-x-3-1/2x^2+x-1

$$\frac { 1 } { 2 x ^ { 2 } - x - 3 } - \frac { 1 } { 2 x ^ { 2 } + x - 1 }$$

Evaluate

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

Short Solution Steps

Factor 2x^{2}-x-3. Factor 2x^{2}+x-1.

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

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

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

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

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

Do the multiplications in 2x-1-\left(2x-3\right).

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

Combine like terms in 2x-1-2x+3.

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

Expand \left(2x-3\right)\left(2x-1\right)\left(x+1\right).

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

Differentiate w.r.t. x

$$-\frac{2\left(\left(6x-2\right)^{2}-19\right)}{3\left(\left(2x-3\right)\left(2x-1\right)\left(x+1\right)\right)^{2}}$$