Simplify 2/(1+i)-3/(1-i)

Problem

2/(1+i)−3/(1−i)

Solution

1. Find a common denominator by multiplying the two denominators together.

Denominator=(1+i)*(1−i)

2. Expand the denominator using the difference of squares and the property i2=−1

(1+i)*(1−i)=1−i2

1−(−1)=2

3. Rewrite the numerators by multiplying each fraction's numerator by the opposite denominator.

(2*(1−i)−3*(1+i))/2

4. Distribute the constants in the numerator.

(2−2*i−3−3*i)/2

5. Combine like terms in the numerator by grouping real and imaginary parts.

((2−3)+(−2*i−3*i))/2

(−1−5*i)/2

6. Split the fraction into its real and imaginary components.

−1/2−5/2*i

Final Answer

2/(1+i)−3/(1−i)=−1/2−5/2*i

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