Which of the following is always true

a,b,c are in A.P and G.P then (a, b, c) can be (0, 0, 0)

If $a,b,c,d,e,ƒ$ are positive real numbers and $dƒ-e^2>0.$ The equations $a{x}^2+2bx+c=0,(a\ne 0)$ and $d{x}^2+2ex+ƒ=0(d\ne 0)$ have a common root, then $\frac{d}{a}=\frac{e}{b}=\frac{ƒ}{c}$ 

One root of the equation $a{x}^2+bx+c=0, (a,b\in R\&c\in R-Q\&a\ne 0)$ in the form of $p+\sqrt{q}$ then other root is $p-\sqrt{q}(p\in z,q\in z^{+})$

Exactly one of the root of the equation $a{x}^2+bx+c=0,(a\ne 0)$ lies in the given interval $(k_1,k_2)$ if $ƒ\left(k_1\right).ƒ\left(k_2\right)>0$