I'm having trouble with a pole and barn paradox problem. The problem is as follows:

A pole vaulter is running with a pole at $ v=\frac{\sqrt3}{2}c $. Her pole has a proper length of $L$. She runs into a barn with proper length $\frac{L}{2}$ with doors on the front and back. When the pole vaulter runs into the barn, a farmer tries to close both front and back doors at the same time, but only for an instant, and then reopens them.

What is the expression for the time interval of the door closings in the pole vaulter's frame?

So I know that $\gamma=2$, so in the pole vaulter's frame, the barn has length $\frac{L}{4}$, and in the farmer's frame, the pole has length $\frac{L}{2}$.

In order to find the time interval, I tried using the spacetime interval. In the farmer's frame, the time interval between the two doors closing is 0. This means that the spacetime interval between the two events is $$ (\Delta s) ^2=-(\frac{L}{2})^2$$

Then, I equate this to the spacetime interval from the pole vaulter's frame. $$(\Delta s) ^2=(c\Delta t)^2-(\Delta x)^2 = (c\Delta t)^2 - (\frac{L}{4})^2 = -(\frac{L}{2})^2$$

But solving for $\Delta t$ gives a complex number, when I should be getting a real solution. Why am I getting this result? Does it have to do with how I am selecting my $\Delta x$ for the pole vaulter?