Solve 1/2(t-1)+t+1/t^2+t-2

$$\frac { 1 } { 2 ( t - 1 ) } + \frac { t + 1 } { t ^ { 2 } + t - 2 }$$

Evaluate

$$\frac{3t+4}{2\left(t-1\right)\left(t+2\right)}$$

Short Solution Steps

Factor t^{2}+t-2.

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

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

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

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

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

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

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

Combine like terms in t+2+2t+2.

$$\frac{3t+4}{2\left(t-1\right)\left(t+2\right)}$$

Expand 2\left(t-1\right)\left(t+2\right).

$$\frac{3t+4}{2t^{2}+2t-4}$$

Expand

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

Short Solution Steps

Factor t^{2}+t-2.

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

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

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

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

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

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

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

Combine like terms in t+2+2t+2.

$$\frac{3t+4}{2\left(t-1\right)\left(t+2\right)}$$

Expand 2\left(t-1\right)\left(t+2\right).

$$\frac{3t+4}{2t^{2}+2t-4}$$