If ${\mathrm{b}}^2\ge 4\mathrm{ac}$ for the equation ${\mathrm{ax}}^4+{\mathrm{bx}}^2+\mathrm{c}=0$, then all the roots of the equation will be real, if 

Put ${\mathrm{x}}^2=\mathrm{y}$.
Then, the given equation can be written as $\mathrm{f}\left(\mathrm{y}\right)={\mathrm{ay}}^2+\mathrm{by}+\mathrm{c}=0 ....\left(\mathrm{i}\right)$
The given equation will have four real roots, i.e., Eq. (i) has two non-negative roots.
Then,       $-\frac{\mathrm{b}}{\mathrm{a}}\ge 0$
                      $\mathrm{af}\left(0\right)\ge 0$
and    ${\mathrm{b}}^2-4\mathrm{ac}\ge 0$            [given]
$\begin{array}{r}\Rightarrow \frac{\mathrm{b}}{\mathrm{a}}\le 0\\ \mathrm{ac}\ge 0\end{array}$
$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}>0,\mathrm{b}<0,\mathrm{c}>0\\ \text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}<0,\mathrm{b}>0,\mathrm{c}<0\end{array}$