Question: How do I find the first 10 roots?

Hi, I'm trying to find the first ten roots of the equation:

sin(a*Pi)*sqrt(4*a^2+3)*cos(sqrt(4*a^2+3)*Pi)+sin(sqrt(4*a^2+3)*Pi)*a*cos(a*Pi) = 0

But fsolve only finds one root, and not necessarily the first one. So I'm having a difficult time getting the first 10 roots without preknowledge of the intervals to search in. (I can obviously plot it, but there are some other parameters which affect this equation and I can't manually find the intervals every time I change them).

I've tried NextZero which failed, and also tried using avoid in fsolve but it still didn't work as intended.

Any ideas how I could do this reliably?


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