Answer
$x = 0.707~A$
Work Step by Step
We can write an expression for the total energy in the system.
$E = \frac{1}{2}kA^2$
When the elastic potential energy equals the kinetic energy, then each form of energy will be equal to $\frac{E}{2}$. We can find $x$ when the elastic potential energy is equal to $\frac{E}{2}$.
$\frac{1}{2}kx^2 = \frac{E}{2}$
$\frac{1}{2}kx^2 = \frac{\frac{1}{2}kA^2}{2}$
$x^2 = \frac{A^2}{2}$
$x = \sqrt{\frac{1}{2}}~A$
$x = 0.707~A$
