A particle moves in a potential region given by U = 8x² -4x + 400 J. Its state of equilibrium will be

To find the state of equilibrium of the particle, we need to determine the position where the net force acting on the particle is zero.

The potential energy function of the particle is given by:
$U=8x^2-4x+400$

The relationship between force $F$ and potential energy $U$ in a conservative force field is given by:
$F=-\frac{dU}{dx}$

For the particle to be in a state of equilibrium, the net force acting on it must be zero:
$F=0$
This implies:
$-\frac{dU}{dx}=0\Rightarrow \frac{dU}{dx}=0$

Now, we differentiate $U$ with respect to $x$:
$\frac{d}{dx}(8x^2-4x+400)=0$
$16x-4=0$

Solving for $x$:
$16x=4$
$x=\frac{4}{16}$
$x=0.25\text{m}$

Therefore, the state of equilibrium of the particle will be at $x=0.25\text{m}$.