Solve 3/a-a-2/3-a/2/3a+2

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

Evaluate

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

Short Solution Steps

Express \frac{\frac{a-2}{3-a}}{2} as a single fraction.

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

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

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

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

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

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

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

Combine like terms in -6a+18-a^{2}+2a.

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

Express \frac{\frac{-4a+18-a^{2}}{2a\left(-a+3\right)}}{3a+2} as a single fraction.

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

Use the distributive property to multiply 2a by -a+3.

$$\frac{-4a+18-a^{2}}{\left(-2a^{2}+6a\right)\left(3a+2\right)}$$

Apply the distributive property by multiplying each term of -2a^{2}+6a by each term of 3a+2.

$$\frac{-4a+18-a^{2}}{-6a^{3}-4a^{2}+18a^{2}+12a}$$

Combine -4a^{2} and 18a^{2} to get 14a^{2}.

$$\frac{-4a+18-a^{2}}{-6a^{3}+14a^{2}+12a}$$

Expand

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

Short Solution Steps

Express \frac{\frac{a-2}{3-a}}{2} as a single fraction.

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

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

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

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

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

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

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

Combine like terms in -6a+18-a^{2}+2a.

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

Express \frac{\frac{-4a+18-a^{2}}{2a\left(-a+3\right)}}{3a+2} as a single fraction.

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

Use the distributive property to multiply 2a by -a+3.

$$\frac{-4a+18-a^{2}}{\left(-2a^{2}+6a\right)\left(3a+2\right)}$$

Apply the distributive property by multiplying each term of -2a^{2}+6a by each term of 3a+2.

$$\frac{-4a+18-a^{2}}{-6a^{3}-4a^{2}+18a^{2}+12a}$$

Combine -4a^{2} and 18a^{2} to get 14a^{2}.

$$\frac{-4a+18-a^{2}}{-6a^{3}+14a^{2}+12a}$$