Analytically solve equation with two normal CDFs

Question

I'm seeking an analytical solution to the equation $$ \Phi(x + c) - \Phi(-x) = \alpha $$ where $\Phi$ is the standard normal cumulative distribution function, $c \in \mathbb{R}$ and $\alpha \in (0, 1)$ are known, and $x > 0$ is unknown. I know it's relatively straightforward to solve using a polynomial expansion for $\Phi$, or by using Newton's method or similar, but I'm curious if a simpler solution is possible.