Solve 5+6i/2-i

$$\frac { 5 + 6 i } { 2 - i }$$

Evaluate

$$\frac{4}{5}+\frac{17}{5}i=0.8+3.4i$$

Solution Steps

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

$$\frac{\left(5+6i\right)\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}$$

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

$$\frac{\left(5+6i\right)\left(2+i\right)}{2^{2}-i^{2}}$$

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

$$\frac{\left(5+6i\right)\left(2+i\right)}{5}$$

Multiply complex numbers 5+6i and 2+i like you multiply binomials.

$$\frac{5\times 2+5i+6i\times 2+6i^{2}}{5}$$

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

$$\frac{5\times 2+5i+6i\times 2+6\left(-1\right)}{5}$$

Do the multiplications in 5\times 2+5i+6i\times 2+6\left(-1\right).

$$\frac{10+5i+12i-6}{5}$$

Combine the real and imaginary parts in 10+5i+12i-6.

$$\frac{10-6+\left(5+12\right)i}{5}$$

Do the additions in 10-6+\left(5+12\right)i.

$$\frac{4+17i}{5}$$

Divide 4+17i by 5 to get \frac{4}{5}+\frac{17}{5}i.

$$\frac{4}{5}+\frac{17}{5}i$$

Real Part

$$\frac{4}{5} = 0.8$$

Solution Steps

Multiply both numerator and denominator of \frac{5+6i}{2-i} by the complex conjugate of the denominator, 2+i.

$$Re(\frac{\left(5+6i\right)\left(2+i\right)}{\left(2-i\right)\left(2+i\right)})$$

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

$$Re(\frac{\left(5+6i\right)\left(2+i\right)}{2^{2}-i^{2}})$$

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

$$Re(\frac{\left(5+6i\right)\left(2+i\right)}{5})$$

Multiply complex numbers 5+6i and 2+i like you multiply binomials.

$$Re(\frac{5\times 2+5i+6i\times 2+6i^{2}}{5})$$

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

$$Re(\frac{5\times 2+5i+6i\times 2+6\left(-1\right)}{5})$$

Do the multiplications in 5\times 2+5i+6i\times 2+6\left(-1\right).

$$Re(\frac{10+5i+12i-6}{5})$$

Combine the real and imaginary parts in 10+5i+12i-6.

$$Re(\frac{10-6+\left(5+12\right)i}{5})$$

Do the additions in 10-6+\left(5+12\right)i.

$$Re(\frac{4+17i}{5})$$

Divide 4+17i by 5 to get \frac{4}{5}+\frac{17}{5}i.

$$Re(\frac{4}{5}+\frac{17}{5}i)$$

The real part of \frac{4}{5}+\frac{17}{5}i is \frac{4}{5}.

$$\frac{4}{5}$$