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Q.
If $a^2+4 b^2+9 c^2+2 a b+6 b c+3 a c \leq 0$ where $a, b, c \in R$, then the number of distinct real roots of $a x^2+b x+c=0$ will be
Complex Numbers and Quadratic Equations
Solution:
$\Theta 2 a^2+8 b^2+18 c^2+4 a b+12 b c+6 a c \leq 0$
$\Rightarrow( a +2 b )^2+(2 b +3 c )^2+(3 c + a )^2 \leq 0 $
$\Rightarrow a +2 b =0 ; 2 b +3 c =0 \text { and } 3 c + a =0$
$\Rightarrow a = b = c =0$
$\therefore$ Equation $ax ^2+ bx + c =0$ will be an identity