Solve 9-2i/sqrt{5+i}

$$\frac { 9 - 2 i } { \sqrt { 5 } + i }$$

Evaluate

$$\sqrt{5}\left(\frac{3}{2}-\frac{1}{3}i\right)+\left(-\frac{1}{3}-\frac{3}{2}i\right)\approx 3.020768633-2.245355992i$$

Solution Steps

Rationalize the denominator of \frac{9-2i}{\sqrt{5}+i} by multiplying numerator and denominator by \sqrt{5}-i.

$$\frac{\left(9-2i\right)\left(\sqrt{5}-i\right)}{\left(\sqrt{5}+i\right)\left(\sqrt{5}-i\right)}$$

Consider \left(\sqrt{5}+i\right)\left(\sqrt{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}.

$$\frac{\left(9-2i\right)\left(\sqrt{5}-i\right)}{\left(\sqrt{5}\right)^{2}-i^{2}}$$

Square \sqrt{5}. Square i.

$$\frac{\left(9-2i\right)\left(\sqrt{5}-i\right)}{5+1}$$

Subtract -1 from 5 to get 6.

$$\frac{\left(9-2i\right)\left(\sqrt{5}-i\right)}{6}$$

Divide \left(9-2i\right)\left(\sqrt{5}-i\right) by 6 to get \left(\frac{3}{2}-\frac{1}{3}i\right)\left(\sqrt{5}-i\right).

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

Use the distributive property to multiply \frac{3}{2}-\frac{1}{3}i by \sqrt{5}-i.

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

Real Part

$$\frac{3 \sqrt{5}}{2} - \frac{1}{3} = 3.020768633$$