Solve 2-5i/1-2i

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

Evaluate

$$\frac{12}{5}-\frac{1}{5}i=2.4-0.2i$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{2+4i-5i+10}{5}$$

Combine the real and imaginary parts in 2+4i-5i+10.

$$\frac{2+10+\left(4-5\right)i}{5}$$

Do the additions in 2+10+\left(4-5\right)i.

$$\frac{12-i}{5}$$

Divide 12-i by 5 to get \frac{12}{5}-\frac{1}{5}i.

$$\frac{12}{5}-\frac{1}{5}i$$

Real Part

$$\frac{12}{5} = 2\frac{2}{5} = 2.4$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{2+4i-5i+10}{5})$$

Combine the real and imaginary parts in 2+4i-5i+10.

$$Re(\frac{2+10+\left(4-5\right)i}{5})$$

Do the additions in 2+10+\left(4-5\right)i.

$$Re(\frac{12-i}{5})$$

Divide 12-i by 5 to get \frac{12}{5}-\frac{1}{5}i.

$$Re(\frac{12}{5}-\frac{1}{5}i)$$

The real part of \frac{12}{5}-\frac{1}{5}i is \frac{12}{5}.

$$\frac{12}{5}$$