If the roots of the equation bx^2 + cx + a = 0 imaginary, then for all real values of x, t…

Question

If the roots of the equation $b{x^2} + cx + a = 0$ imaginary, then for all real values of $x$, the expression $3{b^2}{x^2} + 6bcx + 2{c^2}$ is :

Accepted Answer

Given that roots of the equation $b{x^2} + cx + a = 0$ are imaginary $$\therefore$$ $${c^2} - 4ab < 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$ Let $$y = 3{b^2}{x^2} + 6bc\,x + 2{c^2}$$ $$ \Rightarrow 3{b^2}{x^2} + 6bc\,x + 2{c^2} - y = 0$$ As $x$ is real, $$D \ge 0$$ $$ \Rightarrow 36{b^2}{c^2} - 12{b^2}\left( {2{c^2} - y} \right) \ge 0$$ $$ \Rightarrow 12{b^2}\left( {3{c^2} - 2{c^2} + y} \right) \ge 0$$ $$ \Rightarrow {c^2} + y \ge 0$$ $$ \Rightarrow y \ge - {c^2}$$ But from eqn. $(i),$ ${c^2} < 4ab$ or $- {c^2} > - 4ab$ $$\therefore$$ we get $$y \ge - {c^2} > - 4ab$$ $y > - 4ab$

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