Show that for any eigenvalue we can find a real-valued eigenfunction (for the laplacian)

Question

Let $\Omega \in \mathbb{R}^d$ be an open bounded set with smooth boundary. Consider $$-\bigtriangleup q(x) = \lambda q(x)$$ with either Dirichlet or Neumann boundary conditions $$q(x)=0, x \in \partial \Omega$$or $$\bigtriangledown q(x) \centerdot n(x) = 0, x \in \partial \Omega$$ I need to show that for any eigenvalue, we can find a real-valued eigenfunction. I have shown that the eigenvalues are real by taking the conjugate of the pde, multiplying the conjugate one by $q$ and the non-conjugate one by the conjugate of $q$ and integrating. I've tried subtracting the conjugate equation from the original and integrating but that didn't get me anywhere. The best I can get from doing any integration is that $\frac{q(x)}{\bigtriangleup q(x)}$ is real but that follows immediately from the pde if you know $\lambda$ is real. I've been stuck for a while and don't know what to do. Thanks.