<s>[INST] A biquadratic equation to have at least two real roots Which of the equations have at least two real roots?
\begin{aligned}
x^4-5x^2-36 & = 0 & (1) \\
x^4-13x^2+36 & = 0 & (2) \\
4x^4-10x^2+25 & = 0 & (3) 
\end{aligned}
I wasn't able to notice something clever, so I solved each of the equations. The first one has $2$ real roots, the second one $4$ real roots and the last one does not have real roots. I am pretty sure that the idea behind the problem wasn't solving each of the equations. What can we note to help us solve it faster?
 [/INST] $$  x^4 -13 x^2 + 36 = (x^2 + 6)^2 - 25 x^2 = (x^2 + 5x+6) (x^2 - 5x+6) $$
and both quadratic factors have real roots(positive discriminants).
$$ 4 x^4 -10 x^2 + 25 = (2x^2 + 5)^2 - 30 x^2 = (2x^2 + x \sqrt{30}+5)  (2x^2 - x \sqrt{30}+5) $$
and both quadratic factors have negative discriminants.  OR
$$ 4 x^4 -10 x^2 + 25 = (2x^2 - 5)^2 +10 x^2  $$
is always positive
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