If $c < 0$ and $a x^{2} + b x + c = 0$ does not have any real roots then prove that $9 a + 3 b + c < 0$

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Solution

Let $f (x) = a x^{2} + b x + c = 0$
Given $c < 0$
$f (0) < 0$
Given $f (x)$ has no real roots:
$∴ f (x) < 0 \forall x ϵ R$ & $a < 0$
$f (x) > 0 \forall x ϵ R$ & $a > 0$
If $f (0) < 0$ then $a$ must be negative
and $f (x) < 0 \forall x ϵ R$
$∴ f (3) < 0 \Rightarrow 9 a + 3 b + c < 0$

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