Solve the following equation and check for extraneous solutions. \[ \sqrt[4]{-2 x^{2}+1}=-x \] Enter the solution(s) below. Round your answer to three decimal places. There are four solutions. They are: \[ x_{1}= \] \[ x_{2}= \] \[ x_{3}= \] \[ x_{4}= \] There are two solutions. They are: \[ x_{1}=\quad x_{2}= \] There is one solution. It is: \[ x_{1}= \] There is no solution.

Step 1 ：The given equation is \(\sqrt[4]{-2 x^{2}+1}=-x\).

Step 2 ：Square both sides of the equation to get rid of the fourth root, which gives us \((-2x^2 + 1) = x^2\).

Step 3 ：Simplify this equation to get a quadratic equation in terms of \(x\), which is \(3x^2 + 1 = 0\).

Step 4 ：Solve this quadratic equation to find the possible values of \(x\), which gives us \(x = -\frac{\sqrt{3}}{3}\).

Step 5 ：Substitute these values back into the original equation to check for extraneous solutions. The solution \(x = -\frac{\sqrt{3}}{3}\) is valid.

Step 6 ：Final Answer: There is one solution. It is: \(x_{1}=\boxed{-0.577}\).