<s>[INST] Show that the roots of the equation $x^2-4abx+(a^2+2b^2)^2=0$ are imaginary. Show that the roots of the equation $x^2-4abx+(a^2+2b^2)^2=0$ are imaginary.
My Attempt:
$$x^2-4abx+(a^2+2b^2)^2=0$$
Comparing above equation with $Ax^2+Bx+C=0$, we get
\begin{align}
A&=1 \\
B&=-4ab \\
C&=(a^2+2b^2)^2
\end{align}
Now,
\begin{align}
B^2-4AC
&=(-4ab)^2 - 4\cdot 1\cdot (a^2+2b^2)^2 \\
&=16a^2b^2 - 4(a^4+4a^2b^2+4b^4) \\
&=-4a^4-16b^4. \end{align}
 [/INST] If  $a=b=0$ and the root is $0$.
However, if $(a,b) \neq 0$, then 
the discriminant is equal to $-4(a^4+4b^4)<0$, hence the roots are imaginary.
</s>