Quartic Equation: Distinct real roots

Similar to in my other question , I will find the turning points of the LHS so we can sketch the graph.

Let $f(x)=27x^4-18x^2+8x$ .

$\begin{aligned}f'(x)&=0\\\iff 108x^3-36x+8&=0\\\iff 27x^3-9x+2&=0\\\iff (3x-1)^2(3x+2)&=0\\\iff x=-\frac{2}{3}\textnormal{ or }\frac{1}{3}\end{aligned}$

$f\left (\dfrac{1}{3}\right )=1$ , $f\left (-\dfrac{2}{3}\right )=-8$

Further, $f''(x)=324x^2-36$ and there is a sign change either side of $x=\dfrac{1}{3}$ , so we can conclude that there is a stationary point of inflection at $\left(\dfrac{1}{3},1\right)$ .

The sketch shows there are $\color{#20A900}\boxed{2}$ distinct real roots since, despite the graph of $y=1.01$ coming close to the point of inflection at $x=\dfrac{1}{3}$ , we have shown there are no turning points there so there can only be one intersection, in addition to the one at $x\approx -1$ .