Which are the conditions for a biquadratic equation to have $~4~$ different roots?

Question

Which are the conditions for a biquadratic equation to have $~4~$ different roots in $~\mathbb R~$? I think $~D>0~$, If we have $$t=x^2$$ then $~t>0~$. Is there any other condition?

Answer 1

The biquadratic polynomial $ax^4+bx^2+c$ will have four distinct roots precisely if $at^2+bt+c=0$ has two distinct roots (since distinct numbers can't have identical square roots) and these two roots are $\not= 0$. This happens if and only if $b^2-4ac\not=0$ and $c\not= 0$.