How to find the coefficient of $a$ in a quadratic equation?

Question

If a root of the equation $$3x^2 + 4x + 12a + 9ax = 0$$ is greater than 6, then the correct statement of the coefficient $a$ is:

a) $a = 2$

b) $a> -2$

c) $a = -2$

d) $a <-2$

e) $-2\le a\le 2$

What I did was solve it as if a root was exactly $6$, and I find that $a$ is equal to $2$ ... But the premise says that a root has to be greater than $6$, not equal to $6$ ... and I do not know how find that, for now I would know that $a$ does not have to be $2$ ...

Answer 1

Hint

\begin{align} 3x^2+4x+12a+9ax = 0&\iff 3x^2+(4+9a)x+12a= 0\\ &\iff x=\dfrac{-4-9a\pm \sqrt{(4+9a)^2-144a}}{6}.\end{align} So,

$$\dfrac{-4-9a\pm \sqrt{(4+9a)^2-144a}}{6}\ge 6 \iff -4-9a\pm \sqrt{(4+9a)^2-144a}\ge 36. $$

That is

$$-4-9a\pm \sqrt{16+72a -63a^2}\ge 36$$

So, you can check what is the correct answer.

Answer 2

Hint: $\;3x^2 + 4x + 12a + 9ax = x(3x+4)+3a(3x+4)= (x+3a)(3x+4)\,$.

Answer 3

Using @dxiv hint $(x+3a)(3x+4) = 0$ we have $x = -\frac{4}{3} < 6$ and $x = -3a > 6$ Therefore, $a < -2$