Solve 3+5i/1-2i

$$\frac { 3 + 5 i } { 1 - 2 i }$$

Evaluate

$$-\frac{7}{5}+\frac{11}{5}i=-1.4+2.2i$$

Solution Steps

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

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

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

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

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

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

Multiply complex numbers 3+5i and 1+2i like you multiply binomials.

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

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

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

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

$$\frac{3+6i+5i-10}{5}$$

Combine the real and imaginary parts in 3+6i+5i-10.

$$\frac{3-10+\left(6+5\right)i}{5}$$

Do the additions in 3-10+\left(6+5\right)i.

$$\frac{-7+11i}{5}$$

Divide -7+11i by 5 to get -\frac{7}{5}+\frac{11}{5}i.

$$-\frac{7}{5}+\frac{11}{5}i$$

Real Part

$$-\frac{7}{5} = -1\frac{2}{5} = -1.4$$

Solution Steps

Multiply both numerator and denominator of \frac{3+5i}{1-2i} by the complex conjugate of the denominator, 1+2i.

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

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

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

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

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

Multiply complex numbers 3+5i and 1+2i like you multiply binomials.

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

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

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

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

$$Re(\frac{3+6i+5i-10}{5})$$

Combine the real and imaginary parts in 3+6i+5i-10.

$$Re(\frac{3-10+\left(6+5\right)i}{5})$$

Do the additions in 3-10+\left(6+5\right)i.

$$Re(\frac{-7+11i}{5})$$

Divide -7+11i by 5 to get -\frac{7}{5}+\frac{11}{5}i.

$$Re(-\frac{7}{5}+\frac{11}{5}i)$$

The real part of -\frac{7}{5}+\frac{11}{5}i is -\frac{7}{5}.

$$-\frac{7}{5}$$