I'm trying to find such $a$ so the system has 3 positive solutions: $$3x^2-24x+64=a|x-3|$$ The right answer is $6+2\sqrt{57}; (21\frac13; +\infty )$.
I made a graph and figured out that $a>21\frac13$. The next step was to find where the left side of module function would be a tangent to the parabola. $a=6+2\sqrt{57}$ there. $6+2\sqrt{57} < 21\frac13$. That means that the line with $a=21\frac13$ rises more rapidly.
What I don't understand is why it necessary means that the point of tangency is lower than $(0;64)$ where $a=21\frac13$. (Because if that is true, $x>0$).
I've just started learning to solve problems with parameter. Maybe there's some simple property of function I don't know. Please explain this detail.