How much is x?

16*​x^​6—​24*​x^​4+​8.5*​x^​2—​x-​0.5

x has four values for which y = 0.

I agree, but which are those values?

The real values are:

X1=-0.19928315255971443
X4=-0.8798520812734674
X5=-0.8474547040032134
X6=1.04673785656293

I’d start by squaring both sides:
(4x^3 – 3x) ^2 = (x+1)/2
Expand
16x^6 – 24x^4 + 9x^2 = (x+1)/2
double & collect terms -> polynomial form:
32x^6 – 48x^4 + 18x^2 – x – 1 = 0
(This differs from @Charles‘s equation)
Note that x=1 clearly satisfies the original equation, so that makes an easy test.

I went online to find polynomial root calculators. Here’s the one I used, which gives the following:
———————
The real roots of the polynomial 32×6 – 48×4 + 18×2 – x – 1 are:
X1 = 1
X2 = 0.309016997462
X3 = 0.309017252422
X4 = -0.222520701724
X5 = -0.809015489567
X6 = -0.809015938669