Solve (2+i)/3-i

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

Evaluate

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

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3+i.

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

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

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

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

$$\frac{\left(2+i\right)\left(3+i\right)}{10}$$

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

$$\frac{2\times 3+2i+3i+i^{2}}{10}$$

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

$$\frac{2\times 3+2i+3i-1}{10}$$

Do the multiplications in 2\times 3+2i+3i-1.

$$\frac{6+2i+3i-1}{10}$$

Combine the real and imaginary parts in 6+2i+3i-1.

$$\frac{6-1+\left(2+3\right)i}{10}$$

Do the additions in 6-1+\left(2+3\right)i.

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

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

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

Real Part

$$\frac{1}{2} = 0.5$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(2+i\right)\left(3+i\right)}{10})$$

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

$$Re(\frac{2\times 3+2i+3i+i^{2}}{10})$$

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

$$Re(\frac{2\times 3+2i+3i-1}{10})$$

Do the multiplications in 2\times 3+2i+3i-1.

$$Re(\frac{6+2i+3i-1}{10})$$

Combine the real and imaginary parts in 6+2i+3i-1.

$$Re(\frac{6-1+\left(2+3\right)i}{10})$$

Do the additions in 6-1+\left(2+3\right)i.

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

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

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

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

$$\frac{1}{2}$$