Solve (2/a-2-3/a+2)*4-a^2/10a-a^2

$$( \frac { 2 } { a - 2 } - \frac { 3 } { a + 2 } ) \cdot \frac { 4 - a ^ { 2 } } { 10 a - a ^ { 2 } }$$

Evaluate

$$-\frac{1}{a}$$

Short Solution Steps

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

$$\left(\frac{2\left(a+2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{3\left(a-2\right)}{\left(a-2\right)\left(a+2\right)}\right)\times \frac{4-a^{2}}{10a-a^{2}}$$

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

$$\frac{2\left(a+2\right)-3\left(a-2\right)}{\left(a-2\right)\left(a+2\right)}\times \frac{4-a^{2}}{10a-a^{2}}$$

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

$$\frac{2a+4-3a+6}{\left(a-2\right)\left(a+2\right)}\times \frac{4-a^{2}}{10a-a^{2}}$$

Combine like terms in 2a+4-3a+6.

$$\frac{-a+10}{\left(a-2\right)\left(a+2\right)}\times \frac{4-a^{2}}{10a-a^{2}}$$

Multiply \frac{-a+10}{\left(a-2\right)\left(a+2\right)} times \frac{4-a^{2}}{10a-a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-a+10\right)\left(4-a^{2}\right)}{\left(a-2\right)\left(a+2\right)\left(10a-a^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(a-2\right)\left(-a-2\right)\left(-a+10\right)}{a\left(a-2\right)\left(a+2\right)\left(-a+10\right)}$$

Extract the negative sign in -2-a.

$$\frac{-\left(a-2\right)\left(a+2\right)\left(-a+10\right)}{a\left(a-2\right)\left(a+2\right)\left(-a+10\right)}$$

Cancel out \left(a-2\right)\left(a+2\right)\left(-a+10\right) in both numerator and denominator.

$$\frac{-1}{a}$$

Expand

$$-\frac{1}{a}$$

Short Solution Steps

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

$$\left(\frac{2\left(a+2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{3\left(a-2\right)}{\left(a-2\right)\left(a+2\right)}\right)\times \frac{4-a^{2}}{10a-a^{2}}$$

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

$$\frac{2\left(a+2\right)-3\left(a-2\right)}{\left(a-2\right)\left(a+2\right)}\times \frac{4-a^{2}}{10a-a^{2}}$$

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

$$\frac{2a+4-3a+6}{\left(a-2\right)\left(a+2\right)}\times \frac{4-a^{2}}{10a-a^{2}}$$

Combine like terms in 2a+4-3a+6.

$$\frac{-a+10}{\left(a-2\right)\left(a+2\right)}\times \frac{4-a^{2}}{10a-a^{2}}$$

Multiply \frac{-a+10}{\left(a-2\right)\left(a+2\right)} times \frac{4-a^{2}}{10a-a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-a+10\right)\left(4-a^{2}\right)}{\left(a-2\right)\left(a+2\right)\left(10a-a^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(a-2\right)\left(-a-2\right)\left(-a+10\right)}{a\left(a-2\right)\left(a+2\right)\left(-a+10\right)}$$

Extract the negative sign in -2-a.

$$\frac{-\left(a-2\right)\left(a+2\right)\left(-a+10\right)}{a\left(a-2\right)\left(a+2\right)\left(-a+10\right)}$$

Cancel out \left(a-2\right)\left(a+2\right)\left(-a+10\right) in both numerator and denominator.

$$\frac{-1}{a}$$