Solve 3+i/2-i+2-i/3+i

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

Evaluate

$$\frac{3}{2}+\frac{1}{2}i=1.5+0.5i$$

Solution Steps

Multiply both numerator and denominator of \frac{3+i}{2-i} by the complex conjugate of the denominator, 2+i.

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

Do the multiplications in \frac{\left(3+i\right)\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}.

$$\frac{5+5i}{5}+\frac{2-i}{3+i}$$

Divide 5+5i by 5 to get 1+i.

$$1+i+\frac{2-i}{3+i}$$

Multiply both numerator and denominator of \frac{2-i}{3+i} by the complex conjugate of the denominator, 3-i.

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

Do the multiplications in \frac{\left(2-i\right)\left(3-i\right)}{\left(3+i\right)\left(3-i\right)}.

$$1+i+\frac{5-5i}{10}$$

Divide 5-5i by 10 to get \frac{1}{2}-\frac{1}{2}i.

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

Add 1+i and \frac{1}{2}-\frac{1}{2}i to get \frac{3}{2}+\frac{1}{2}i.

$$\frac{3}{2}+\frac{1}{2}i$$

Real Part

$$\frac{3}{2} = 1\frac{1}{2} = 1.5$$

Solution Steps

Multiply both numerator and denominator of \frac{3+i}{2-i} by the complex conjugate of the denominator, 2+i.

$$Re(\frac{\left(3+i\right)\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}+\frac{2-i}{3+i})$$

Do the multiplications in \frac{\left(3+i\right)\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}.

$$Re(\frac{5+5i}{5}+\frac{2-i}{3+i})$$

Divide 5+5i by 5 to get 1+i.

$$Re(1+i+\frac{2-i}{3+i})$$

Multiply both numerator and denominator of \frac{2-i}{3+i} by the complex conjugate of the denominator, 3-i.

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

Do the multiplications in \frac{\left(2-i\right)\left(3-i\right)}{\left(3+i\right)\left(3-i\right)}.

$$Re(1+i+\frac{5-5i}{10})$$

Divide 5-5i by 10 to get \frac{1}{2}-\frac{1}{2}i.

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

Add 1+i and \frac{1}{2}-\frac{1}{2}i to get \frac{3}{2}+\frac{1}{2}i.

$$Re(\frac{3}{2}+\frac{1}{2}i)$$

The real part of \frac{3}{2}+\frac{1}{2}i is \frac{3}{2}.

$$\frac{3}{2}$$