Solve 3+4i/1-2i

$$\frac { 3 + 4 i } { 1 - 2 i }$$

Evaluate

$$-1+2i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1+2i.

$$\frac{\left(3+4i\right)\left(1+2i\right)}{\left(1-2i\right)\left(1+2i\right)}$$

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

$$\frac{\left(3+4i\right)\left(1+2i\right)}{1^{2}-2^{2}i^{2}}$$

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

$$\frac{\left(3+4i\right)\left(1+2i\right)}{5}$$

Multiply complex numbers 3+4i and 1+2i like you multiply binomials.

$$\frac{3\times 1+3\times \left(2i\right)+4i\times 1+4\times 2i^{2}}{5}$$

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

$$\frac{3\times 1+3\times \left(2i\right)+4i\times 1+4\times 2\left(-1\right)}{5}$$

Do the multiplications in 3\times 1+3\times \left(2i\right)+4i\times 1+4\times 2\left(-1\right).

$$\frac{3+6i+4i-8}{5}$$

Combine the real and imaginary parts in 3+6i+4i-8.

$$\frac{3-8+\left(6+4\right)i}{5}$$

Do the additions in 3-8+\left(6+4\right)i.

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

Divide -5+10i by 5 to get -1+2i.

$$-1+2i$$

Real Part

$$-1$$

Solution Steps

Multiply both numerator and denominator of \frac{3+4i}{1-2i} by the complex conjugate of the denominator, 1+2i.

$$Re(\frac{\left(3+4i\right)\left(1+2i\right)}{\left(1-2i\right)\left(1+2i\right)})$$

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

$$Re(\frac{\left(3+4i\right)\left(1+2i\right)}{1^{2}-2^{2}i^{2}})$$

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

$$Re(\frac{\left(3+4i\right)\left(1+2i\right)}{5})$$

Multiply complex numbers 3+4i and 1+2i like you multiply binomials.

$$Re(\frac{3\times 1+3\times \left(2i\right)+4i\times 1+4\times 2i^{2}}{5})$$

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

$$Re(\frac{3\times 1+3\times \left(2i\right)+4i\times 1+4\times 2\left(-1\right)}{5})$$

Do the multiplications in 3\times 1+3\times \left(2i\right)+4i\times 1+4\times 2\left(-1\right).

$$Re(\frac{3+6i+4i-8}{5})$$

Combine the real and imaginary parts in 3+6i+4i-8.

$$Re(\frac{3-8+\left(6+4\right)i}{5})$$

Do the additions in 3-8+\left(6+4\right)i.

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

Divide -5+10i by 5 to get -1+2i.

$$Re(-1+2i)$$

The real part of -1+2i is -1.

$$-1$$