<s>[INST] Find the range of $m$ for which atleast one of the following equations $ax^2+bx+cm=0,bx^2+cx+am=0,cx^2+ax+bm=0$ have real roots. Let $a,b,c$ and $m\in R^+$.Find the range of $m$ for which atleast one of the following equations $ax^2+bx+cm=0,bx^2+cx+am=0,cx^2+ax+bm=0$ have real roots.

Either one or two or all of the three equations $ax^2+bx+cm=0,bx^2+cx+am=0,cx^2+ax+bm=0$ have the real roots.
But i do not know how to find the range of $m.$Any help will be appreciated.Thanks.
 [/INST] Claim: $$m \leq \dfrac{1}{4}$$
For if all of them do not have real roots then : $a^2-4bcm < 0, b^2-4acm < 0, c^2-4abm < 0 \Rightarrow a^2+b^2+c^2 < 4m(ab+bc+ca) \Rightarrow m > \dfrac{a^2+b^2+c^2}{4ab+4bc+4ca} \geq \dfrac{ab+bc+ca}{4ab+4bc+4ca} = \dfrac{1}{4}$. Thus if $m \leq \dfrac{1}{4}$, then at least one equation has real roots.
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