Solve 2a/a^2+2a-3+(a-3/2a*a^2/a^2-a)=

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}}{a^{2}-a}.

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

Cancel out a in both numerator and denominator.

$$\frac{2a}{a^{2}+2a-3}+\frac{a-3}{2a}\times \frac{a}{a-1}$$

Multiply \frac{a-3}{2a} times \frac{a}{a-1} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out a in both numerator and denominator.

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

Factor a^{2}+2a-3.

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

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

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

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

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

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

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

Combine like terms in 4a+a^{2}+3a-3a-9.

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

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

$$\frac{4a+a^{2}-9}{2a^{2}+4a-6}$$

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}}{a^{2}-a}.

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

Cancel out a in both numerator and denominator.

$$\frac{2a}{a^{2}+2a-3}+\frac{a-3}{2a}\times \frac{a}{a-1}$$

Multiply \frac{a-3}{2a} times \frac{a}{a-1} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out a in both numerator and denominator.

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

Factor a^{2}+2a-3.

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

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

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

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

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

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

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

Combine like terms in 4a+a^{2}+3a-3a-9.

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

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

$$\frac{4a+a^{2}-9}{2a^{2}+4a-6}$$