Solve 1/2a-3-2/3-2a+18/9-4a^2

$$\frac{ 1 }{ 2a-3 } - \frac{ 2 }{ 3-2a } + \frac{ 18 }{ 9-4 { a }^{ 2 } }$$

Evaluate

$$\frac{3}{2a+3}$$

Short Solution Steps

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

$$\frac{1}{2a-3}-\frac{2\left(-1\right)}{2a-3}+\frac{18}{9-4a^{2}}$$

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

$$\frac{1-2\left(-1\right)}{2a-3}+\frac{18}{9-4a^{2}}$$

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

$$\frac{1+2}{2a-3}+\frac{18}{9-4a^{2}}$$

Do the calculations in 1+2.

$$\frac{3}{2a-3}+\frac{18}{9-4a^{2}}$$

Factor 9-4a^{2}.

$$\frac{3}{2a-3}+\frac{18}{\left(-2a-3\right)\left(2a-3\right)}$$

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

$$\frac{3\left(-2a-3\right)}{\left(-2a-3\right)\left(2a-3\right)}+\frac{18}{\left(-2a-3\right)\left(2a-3\right)}$$

Since \frac{3\left(-2a-3\right)}{\left(-2a-3\right)\left(2a-3\right)} and \frac{18}{\left(-2a-3\right)\left(2a-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{3\left(-2a-3\right)+18}{\left(-2a-3\right)\left(2a-3\right)}$$

Do the multiplications in 3\left(-2a-3\right)+18.

$$\frac{-6a-9+18}{\left(-2a-3\right)\left(2a-3\right)}$$

Combine like terms in -6a-9+18.

$$\frac{-6a+9}{\left(-2a-3\right)\left(2a-3\right)}$$

Factor the expressions that are not already factored in \frac{-6a+9}{\left(-2a-3\right)\left(2a-3\right)}.

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

Extract the negative sign in 3-2a.

$$\frac{-3\left(2a-3\right)}{\left(-2a-3\right)\left(2a-3\right)}$$

Cancel out 2a-3 in both numerator and denominator.

$$\frac{-3}{-2a-3}$$

Differentiate w.r.t. a

$$-\frac{6}{\left(2a+3\right)^{2}}$$