How to calculate full width half max of this curve

Question

I'm trying to calculate the full width half max of this function and I keep receiving non-sense answers. The function is $F(ω)=A_0 (\frac{\frac{1}{τ}}{(\frac{1}{τ^2})+(ω_0 - ω)^2})$ and I then have calculated $F(ω_{1/2}) = \frac{A_0 τ}{2} = A_0 (\frac{\frac{1}{τ}}{(\frac{1}{τ^2})+(ω_0 - ω_{1/2})^2})$ to be $ω_{1/2} = ω_0 \pm \frac{1}{\sqrt{2}τ}$ which is non-sense because the units don't make sense.