Answer
$$\pm \frac{A}{\sqrt{2}}$$
Work Step by Step
From the stated conditions, we see that the potential energy is half of the total energy.
$$E_{pot} =\frac{1}{2}E_{total}$$
When the mass is at an endpoint, x = A, the potential energy is the total energy.
$$\frac{1}{2}kx^2=\frac{1}{2}(\frac{1}{2}kA^2)$$
$$x=\pm \frac{A}{\sqrt{2}}$$
