Prove atleast one of the equations $x^2+b_1x+c_1=0$ and $x^2+b_2x+c_2=0$ has real roots if $b_1b_2=2(c_1+c_2)$ 
If $b_1b_2=2(c_1+c_2)$, then prove that atleast one of the equations $x^2+b_1x+c_1=0$ and $x^2+b_2x+c_2=0$ has real roots.

$$
\Delta_1.\Delta_2=(b_1^2-4c_1)(b_2^2-4c_2)=b_1^2b_2^2-4b_2^2c_1-4b_1^2c_2+16c_1c_2\\
=4(c_1+c_2)^2-4b_2^2c_1-4b_1^2c_2+16c_1c_2\\
=4c^2_1+4c_2^2+8c_1c_2-4b_2^2c_1-4b_1^2c_2+16c_1c_2\\
=4\big[c^2_1+c_2^2+2c_1c_2-b_2^2c_1-b_1^2c_2+4c_1c_2\big]\\
=4\big[c^2_1+c_2^2+6c_1c_2-b_2^2c_1-b_1^2c_2\big]
$$
Why am I not reaching any conclusion from the product of the determinants ?

We can assume both $c_1,c_2$ greater than zero.
$b_1b_2=2(c_1+c_2) = 4\dfrac{(c_1+c_2)}{2}$
$b^2_1b^2_2=16\dfrac{(c_1+c_2)^2}{4}\geq 16c_1c_2$
$b^2_1b^2_2 \geq 4c_14c_2$
So if $b^2_1 \lt 4c_1$ , then $b^2_2 \gt 4c_2$ and vice versa.
