Question:

if the complex number $\displaystyle x + iy $ denoted by z then the complex conjugate number $\displaystyle x-iy $denoted by $\displaystyle a*$.

a) Express $\displaystyle \left |\ z^* \right | $ and $\displaystyle arg(z^*) $ in terms of $\displaystyle \left |\ z \right | $ and $\displaystyle arg(z) $

b)if a, b, and c are real numbers prove that if $\displaystyle az^2+bz+c=0$

then $\displaystyle a(z^*)^2 + b{z^*}+ c=0 $

c)if p and q are complex numbers and $\displaystyle q \neq 0 $, prove that $\displaystyle \left ( \frac{p}{q} \right )^{*}=\frac{p^*}{q^*} $

My attempt

a) $\displaystyle \left |\ z^* \right | = \left |\ x-iy \right | = \sqrt{x^2+(-y)^2} = \sqrt{x^2+y^2} $

therefore $\displaystyle \left |\ z^* \right | = \left |\ z \right | = \sqrt{x^2+y^2} $

$\displaystyle arg(z) = \Theta $ and $\displaystyle arg(z^*) = - \Theta$

therefore $\displaystyle arg(z^*) = - arg(z) $

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) $\displaystyle a(z^*)^2 + b{z^*}+ c=0 $[

sub $\displaystyle z^* = x-iy $ into the equation

$\displaystyle a(x-iy)^2+b(x-iy) +c=0 $

$\displaystyle a(x^2-y^2)+bx+c -iy(2ax-b)=0 $

I am stuck here

plus I dont know how to approach part c please can some one help me thank you