Question
If the roots of the equation $$b{x^2} + cx + a = 0$$ be imaginary, then for all real values of $$x,$$ the expression $$3{b^2}{x^2} + 6bcx + 2{c^2}$$ is
A.
less than $$4ab$$
B.
greater than $$- 4ab$$
C.
less than $$- 4ab$$
D.
greater than $$4ab$$
Answer :
greater than $$- 4ab$$
Solution :
Given that roots of the equation
$$\eqalign{
& b{x^2} + cx + a = 0\,{\text{are}}\,{\text{imaginary}} \cr
& \therefore \,\,\,\,\,{c^2} - 4ab < 0\,\,\,\,.....\left( {\text{i}} \right) \cr
& {\text{Let}}\,y = 3{b^2}{x^2} + 6bcx + 2{c^2} \cr
& \Rightarrow \,\,3{b^2}{x^2} + 6bcx + 2{c^2} - y = 0 \cr
& {\text{As}}\,x\,{\text{is}}\,{\text{real}},D \geqslant 0 \cr
& \Rightarrow \,\,\,36{b^2}{c^2} - 12{b^2}\left( {2{c^2} - y} \right) \geqslant 0 \cr
& \Rightarrow \,\,\,12{b^2}\left( {3{c^2} - 2{c^2} + y} \right) \geqslant 0 \cr
& \Rightarrow \,\,\,{c^2} + y \geqslant 0 \cr
& \Rightarrow \,\,y \geqslant - {c^2} \cr
& {\text{But}}\,{\text{from}}\,{\text{eqn}}{\text{.}}\,\left( {\text{i}} \right),\,{c^2} < 4ab\,\,{\text{or}}\,\, - {c^2} > - 4ab \cr
& \therefore \,\,\,{\text{we}}\,{\text{get}}\,y \geqslant - {c^2} > - 4ab \cr
& \Rightarrow \,\,y > - 4ab \cr} $$