If the roots of bx^2+cx+a=0 are imaginary, then for all real values of x, the expression 3…

Question

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

Accepted Answer

Imaginary roots give $c^2-4ab<0$. Let $y=3b^2x^2+6bcx+2c^2$; for $x$ real, requiring $D\ge0$ (as a quadratic in $x$) gives $y\ge-c^2$. Since $c^2<4ab$, $-c^2>-4ab$, so $y\ge-c^2>-4ab \implies y>-4ab.$

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