<s>[INST] Find maximum value of $|M|$ Let z1 and z2 be two complex numbers such that $z_1 ^2 -4z_2 =16+i20$ . If a and b are roots of $x^2+z_1 x+z_2+M=0$(Where M is a complex number) and $|(a-b)^2|=28$ . then we have to find the minimum and maximum value of $|M|$
My try 
$(a+b)^2 -4ab =28$
$z_1 ^2 -4z_2 -4M =28$
from this I got $|M|=5$
But I could not understand how to find minimum and maximum value .
 [/INST] $$\begin{array}{rcl}
|(a-b)^2| &=& 28 \\
(a-b)^2 &=& 28e^{i\theta} \\
(a+b)^2 - 4ab &=& 28e^{i\theta} \\
z_1^2 - 4z_2 - 4M &=& 28e^{i\theta} \\
16+20i - 4M &=& 28e^{i\theta} \\
4M &=& 16 + 20i - 28e^{i\theta} \\
M &=& 4 + 5i - 7e^{i\theta} \\
\end{array}$$
Thus, $M$ is a circle of radius $7$ with center $4+5i$.
$$\begin{array}{rcccl}
||4+5i| - |7e^{i\theta}|| &\le& |M| &\le& |4+5i| + |7e^{i\theta}| \\
|\sqrt{41} - 7| &\le& |M| &\le& \sqrt{41}+7 \\
7-\sqrt{41} &\le& |M| &\le& \sqrt{41}+7 \\
\end{array}$$
</s>