[Math] If $ax^2+bx+6=0$ doesn’t have $2$ distinct real roots, then find the least value of $ (3a+b)$

quadratics

If $ax^2+bx+6=0$ doesn't have $2$ distinct real roots, then find the least value of $(3a+b)$
$a,b\in \mathbb R$

Any hint for this question?

Best Answer

Picking up from $b^2 \leq 24a$, we have $b^2 + 8b \leq 8(3a + b)$ or $$\frac{1}{8}\left( b^2 + 8b \right) \leq 3a + b$$

Now what is the minimum value of the LHS?

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