Solve 3+2i/2+i=

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

Evaluate

$$\frac{8}{5}+\frac{1}{5}i=1.6+0.2i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 2-i.

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

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

By definition, i^{2} is -1. Calculate the denominator.

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

Multiply complex numbers 3+2i and 2-i like you multiply binomials.

$$\frac{3\times 2+3\left(-i\right)+2i\times 2+2\left(-1\right)i^{2}}{5}$$

By definition, i^{2} is -1.

$$\frac{3\times 2+3\left(-i\right)+2i\times 2+2\left(-1\right)\left(-1\right)}{5}$$

Do the multiplications in 3\times 2+3\left(-i\right)+2i\times 2+2\left(-1\right)\left(-1\right).

$$\frac{6-3i+4i+2}{5}$$

Combine the real and imaginary parts in 6-3i+4i+2.

$$\frac{6+2+\left(-3+4\right)i}{5}$$

Do the additions in 6+2+\left(-3+4\right)i.

$$\frac{8+i}{5}$$

Divide 8+i by 5 to get \frac{8}{5}+\frac{1}{5}i.

$$\frac{8}{5}+\frac{1}{5}i$$

Real Part

$$\frac{8}{5} = 1\frac{3}{5} = 1.6$$

Solution Steps

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

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

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

By definition, i^{2} is -1. Calculate the denominator.

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

Multiply complex numbers 3+2i and 2-i like you multiply binomials.

$$Re(\frac{3\times 2+3\left(-i\right)+2i\times 2+2\left(-1\right)i^{2}}{5})$$

By definition, i^{2} is -1.

$$Re(\frac{3\times 2+3\left(-i\right)+2i\times 2+2\left(-1\right)\left(-1\right)}{5})$$

Do the multiplications in 3\times 2+3\left(-i\right)+2i\times 2+2\left(-1\right)\left(-1\right).

$$Re(\frac{6-3i+4i+2}{5})$$

Combine the real and imaginary parts in 6-3i+4i+2.

$$Re(\frac{6+2+\left(-3+4\right)i}{5})$$

Do the additions in 6+2+\left(-3+4\right)i.

$$Re(\frac{8+i}{5})$$

Divide 8+i by 5 to get \frac{8}{5}+\frac{1}{5}i.

$$Re(\frac{8}{5}+\frac{1}{5}i)$$

The real part of \frac{8}{5}+\frac{1}{5}i is \frac{8}{5}.

$$\frac{8}{5}$$