The range of values of a, so that all the roots of the equation are real and distinct, belong to

Question

The range of values of a, so that all the roots of the equation $$2x^3-3x^2-12x+a=0$$ are real and distinct, belong to

Answer 1

hint: Write your equation in the form $$a=-2x^3+3x^2+12x$$ and use for $$h(x)=-2x^3+3x^2+12x$$ calculus.

Answer 2

Hint:

The function $f(x)=2x^3-3x^2-12x$ has a local maximum $M$ and a local minimum $m$. The equation $f(x)+a=0$ has three distinct real roots if and only if $-a\in (m,M)$.

Answer 3

$f(x)=0$ has good number of real roots if $f_{min}<0$ ab\nd $f_{max}>0$. the given function has min at $x=2$ and max at $x=-1$.So for real roots$ f(2)f(-1)<0$ This gives $(a+7)(a-20)<0 \Rightarrow -7 < a<20$ for three real distinctroots.