We need to solve the following equation for l.
$\frac{n_1 - \ell n_2 + \ell ^2 n_3}{\sqrt{d_1 - \ell d_2 + \ell ^2d_3 - \ell ^3d_4 + \ell ^4d_5}} - \cos{a_0} = 0$
We have already tried linearisation of our formulas so that the above equation would look like this:
$\frac{c_1 - \ell c_2}{\sqrt{c_3 - \ell c_4 + \ell^2c_5}} - \cos{a_0} = 0$
However the result of the second equation is not accurate enough for our calculation. For the second equation we were able to solve with it mathematica. However the second equation could not be solved by mathematica.
What approaches are possible to get the solutions or good approximations of the first equation?