If `a,b, c in R^(+) and 2b=a +c` then check the nature of the roost of `ax^(2) +2bx+c=0`

If "log"(1)/(bc), "log"(1)/(ac) and "log"(1)/(ab) are in A.P, then check the nature of roots of 4ax^(2)+4bx+c=0

Find the derivative of 1/(ax^2 + bx + c )

Let a, b, c gt 0 . Then prove that both the roots of the equation ax^(2)+bx+c=0 has negative real parts.

The determinant Delta=|(a,b,ax+b),(b,c,bx+c),(ax+b,bx+c,c)| is equal to zero, if a)a,b,c are in A.P. b)a,b,c are in G.P. c)a,b,c are in H.P. d) ax^(2) + bx + c = 0