if quadratic $x^2-(a+b+c)x+ab+bc+ca=0$ has complex roots, Then prove that $\sqrt{a}$,$\sqrt{b}$ and $\sqrt{c}$ are sides of triangle 
The following quadratic equation $$x^2-(a+b+c)x+ab+bc+ca=0$$ has complex roots.
Prove that $\sqrt{a}$,$\sqrt{b}$ and $\sqrt{c}$ are sides-lengths of triangle, where $a,b,c \in \mathbb{R^+}$.

Since the quadratic has complex roots, we have Discriminant negative. So
$$(a+b+c)^2 \lt 4ab+4bc+4ca$$ $\implies$
$$a^2+b^2+c^2 \lt 2(ab+bc+ca)$$ $\implies$
$$c^2-2c(a+b)+(a-b)^2 \lt 0$$ $\implies$
$$(a+c-b)^2 \lt 4ac$$ $\implies$
we get
$$a+c-b \lt 2 \sqrt{ac}$$
any clue further?

From the given we obtain:
$$\sum_{cyc}(2ab-a^2)>0$$ or
$$(\sqrt{a}+\sqrt{b}+\sqrt{c})\prod_{cyc}(\sqrt{a}+\sqrt{b}-\sqrt{c})>0$$
because 
\begin{align}
\sum_{cyc}(2ab-a^2)&=4ab-(a+b-c)^2\\
&=\left(2\sqrt{ab}\right)^2-(a+b-c)^2\\
&=(2\sqrt{ab}-a-b+c)(2\sqrt{ab}+a+b-c)\\
&=(c-(\sqrt{a}-\sqrt{b})^2)((\sqrt{a}+\sqrt{b})^2-c)\\
&=(\sqrt{c}-\sqrt{a}+\sqrt{b})(\sqrt{c}+\sqrt{a}-\sqrt{b})(\sqrt{a}+\sqrt{b}-\sqrt{c})(\sqrt{a}+\sqrt{b}+\sqrt{c}).
\end{align}
Let $a\geq b\geq c$.
Hence, $\sqrt{b}+\sqrt{c}-\sqrt{a}>0$ and rest multipliers are also positives. 
Done!
