A4. The equations $|x|^{2}-3|x|+2=0$ and $x^{4}-a x^{2}+4=0$ have the same roots. Determine the value of $a$.
Solution:
For $x \geq 0$, equations $|x|^{2}-3|x|+2=0$ becomes $x^{2}-3 x+2=0$, which has roots $x=1$ and $x=2$. For $x<0$, the equations $|x|^{2}-3|x|+2=0$ becomes $x^{2}+3 x+2=0$, which has roots $x=-1$ and $x=-2$. Thus, $|x|^{2}-3|x|+2=0$ has roots $x= \pm 1, \pm 2$.
A polynomial equation with the same roots has the form: $(x-1)(x+1)(x-2)(x+2)=0$. Rewriting the left hand side as $\left(x^{2}-1\right)\left(x^{2}-4\right)=x^{4}-5 x^{2}+4$ and comparing with $x^{4}-a x^{2}+4=0$, we get $a=5$.
Alternatively, we can substitute $x= \pm 1, \pm 2$ in $x^{4}-a x^{2}+4=0$ to get either $1-a+4=0$ or $16-4 a+4=0$. In both cases we have $a=5$.
Answer: 5 .