If `a ,b ,c` are nonzero real numbers and `a z^2+b z+c+i=0` has purely imaginary roots, then prove that `a=b^2c dot`

if ax^2+bx+c = 0 has imaginary roots and a+c lt b then prove that 4a+c lt 2b

If a and c are odd prime numbers and a x^2+b x+c=0 has rational roots , where b in I , prove that one root of the equation will be independent of a ,b ,c dot

If a ,b ,c are non zero complex numbers of equal modlus and satisfy a z^2+b z+c=0, hen prove that (sqrt(5)-1)//2lt=|z|lt=(sqrt(5)+1)//2.

If x is real and the roots of the equation a x^2+b x+c=0 are imaginary, then prove tat a^2x^2+a b x+a c is always positive.

Column I, Column II If a ,b ,c ,a n dd are four zero real numbers such that (d+a-b)^2+(d+b-c)^2=0 and he root of the equation a(b-c)x^2+b(c-a)x=c(a-b)=0 are real and equal, then, p. a+b+c=0 If the roots the equation (a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0 are real and equal, then, q. a ,b ,ca r einAdotPdot If the equation a x^2+b x+c=0a n dx^3-3x^2+3x-0 have a common real root, then, r. a ,b ,ca r einGdotPdot Let a ,b ,c be positive real number such that the expression b x^2+(sqrt((a+b)^2+b^2))x+(a+c) is non-negative AAx in R , then, s. a ,b ,ca r einHdotPdot

Consider a quadratic equation az^2 + bz + c =0 where a, b, c, are complex numbers. Find the condition that the equation has (i) one purely imaginary root (ii) one purely real root (iii) two purely imaginary roots (iv) two purely real roots

Let a ,b and c be any three nonzero complex number. If |z|=1 and' z ' satisfies the equation a z^2+b z+c=0, prove that a .bar a = c .bar c and |a||b|= sqrt(a c( bar b )^2)

If a , b , c are non-zero real numbers and if the system of equations (a-1)x=y=z (b-1)y=z+x (c-1)z=x+y has a non-trivial solution, then prove that a b+b c+c a=a b c

If a ,b are complex numbers and one of the roots of the equation x^2+a x+b=0 is purely real, whereas the other is purely imaginary, prove that a^2- (bar a)^2=4b .

If a, b and c are positive real and a = 2b + 3c , then the equation ax^(2) + bx + c = 0 has real roots for