Solve 2i/2+i+5/2-i

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

Evaluate

$$\frac{12}{5}+\frac{9}{5}i=2.4+1.8i$$

Solution Steps

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

$$\frac{2i\left(2-i\right)}{\left(2+i\right)\left(2-i\right)}+\frac{5}{2-i}$$

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

$$\frac{2+4i}{5}+\frac{5}{2-i}$$

Divide 2+4i by 5 to get \frac{2}{5}+\frac{4}{5}i.

$$\frac{2}{5}+\frac{4}{5}i+\frac{5}{2-i}$$

Multiply both numerator and denominator of \frac{5}{2-i} by the complex conjugate of the denominator, 2+i.

$$\frac{2}{5}+\frac{4}{5}i+\frac{5\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}$$

Do the multiplications in \frac{5\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}.

$$\frac{2}{5}+\frac{4}{5}i+\frac{10+5i}{5}$$

Divide 10+5i by 5 to get 2+i.

$$\frac{2}{5}+\frac{4}{5}i+\left(2+i\right)$$

Add \frac{2}{5}+\frac{4}{5}i and 2+i to get \frac{12}{5}+\frac{9}{5}i.

$$\frac{12}{5}+\frac{9}{5}i$$

Real Part

$$\frac{12}{5} = 2\frac{2}{5} = 2.4$$

Solution Steps

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

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

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

$$Re(\frac{2+4i}{5}+\frac{5}{2-i})$$

Divide 2+4i by 5 to get \frac{2}{5}+\frac{4}{5}i.

$$Re(\frac{2}{5}+\frac{4}{5}i+\frac{5}{2-i})$$

Multiply both numerator and denominator of \frac{5}{2-i} by the complex conjugate of the denominator, 2+i.

$$Re(\frac{2}{5}+\frac{4}{5}i+\frac{5\left(2+i\right)}{\left(2-i\right)\left(2+i\right)})$$

Do the multiplications in \frac{5\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}.

$$Re(\frac{2}{5}+\frac{4}{5}i+\frac{10+5i}{5})$$

Divide 10+5i by 5 to get 2+i.

$$Re(\frac{2}{5}+\frac{4}{5}i+\left(2+i\right))$$

Add \frac{2}{5}+\frac{4}{5}i and 2+i to get \frac{12}{5}+\frac{9}{5}i.

$$Re(\frac{12}{5}+\frac{9}{5}i)$$

The real part of \frac{12}{5}+\frac{9}{5}i is \frac{12}{5}.

$$\frac{12}{5}$$