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Q. Expression $\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-i \sqrt{2})}$ in $a+i b$ form is

Complex Numbers and Quadratic Equations

Solution:

$\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2 i})-(\sqrt{3}-i \sqrt{2})}$
$=\frac{3^2-(i \sqrt{5})^2}{\sqrt{3}+\sqrt{2} i-\sqrt{3}+i \sqrt{2}} \left[\because\left(z_1-z_2\right)\left(z_1+z_2\right)-z_1^2-z_2^2\right]$
$=\frac{9+5}{2 \sqrt{2} i}$
$=\frac{14 i^4}{2 \sqrt{2} i}$$\left(\because i^4=1\right)$
$=\frac{7}{\sqrt{2}} i^3$
$=\frac{-7 \sqrt{2}}{2} i=0-\frac{7 \sqrt{2} i}{2}$