If the roots of the equation bx^2 + cx + a = | vedclass

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(A) Given that the roots of $bx^2 + cx + a = 0$ are imaginary,the discriminant $D < 0$.$D = c^2 - 4ba < 0$,which implies $c^2 < 4ab$.Let $f(x) = 3b^2x

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greater than $-4ab$

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(A) Given that the roots of $bx^2 + cx + a = 0$ are imaginary,the discriminant $D $D = c^2 - 4ba Let $f(x) = 3b^2x^2 + 6bcx + 2c^2$.We can rewrite $f(x)$ by completing the square:$f(x) = 3(b^2x^2 + 2bcx) + 2c^2$$f(x) = 3(bx + c)^2 - 3c^2 + 2c^2$$f(x) = 3(bx + c)^2 - c^2$.Since $(bx + c)^2 \ge 0$,the minimum value of $f(x)$ is $-c^2$.From $c^2  -4ab$.Therefore,$f(x) = 3(bx + c)^2 - c^2 \ge -c^2 > -4ab$.Thus,$f(x) > -4ab$ for all real values of $x$.

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

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