Solve 3+2i/5-i

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

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, 5+i.

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

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

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

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

$$\frac{\left(3+2i\right)\left(5+i\right)}{26}$$

Multiply complex numbers 3+2i and 5+i like you multiply binomials.

$$\frac{3\times 5+3i+2i\times 5+2i^{2}}{26}$$

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

$$\frac{3\times 5+3i+2i\times 5+2\left(-1\right)}{26}$$

Do the multiplications in 3\times 5+3i+2i\times 5+2\left(-1\right).

$$\frac{15+3i+10i-2}{26}$$

Combine the real and imaginary parts in 15+3i+10i-2.

$$\frac{15-2+\left(3+10\right)i}{26}$$

Do the additions in 15-2+\left(3+10\right)i.

$$\frac{13+13i}{26}$$

Divide 13+13i by 26 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{3+2i}{5-i} by the complex conjugate of the denominator, 5+i.

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

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

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

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

$$Re(\frac{\left(3+2i\right)\left(5+i\right)}{26})$$

Multiply complex numbers 3+2i and 5+i like you multiply binomials.

$$Re(\frac{3\times 5+3i+2i\times 5+2i^{2}}{26})$$

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

$$Re(\frac{3\times 5+3i+2i\times 5+2\left(-1\right)}{26})$$

Do the multiplications in 3\times 5+3i+2i\times 5+2\left(-1\right).

$$Re(\frac{15+3i+10i-2}{26})$$

Combine the real and imaginary parts in 15+3i+10i-2.

$$Re(\frac{15-2+\left(3+10\right)i}{26})$$

Do the additions in 15-2+\left(3+10\right)i.

$$Re(\frac{13+13i}{26})$$

Divide 13+13i by 26 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}.

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