How to pick out the largest root of an equation?

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First, here is a classical way to get solutions of you equation:

sage: a = 5
sage: p = x^2 - 7*a*x + 5
sage: p.solve(x)
[x == -1/2*sqrt(1205) + 35/2, x == 1/2*sqrt(1205) + 35/2]

So, you have a list of solutions. Each solution is of the form x == -1/2*sqrt(1205) + 35/2 which is a symbolic expression. You can get the right hand side of such an equality with the rhs() method:

sage: [s.rhs() for s in p.solve(x)]
[-1/2*sqrt(1205) + 35/2, 1/2*sqrt(1205) + 35/2]

Then, you can take the maximal element of this list:

sage: max([s.rhs() for s in p.solve(x)])
1/2*sqrt(1205) + 35/2

Alternatively, instead of getting solutions as symbolic expressions, you can get them as Python dictionaries:

sage: p.solve(x, solution_dict=True)
[{x: -1/2*sqrt(1205) + 35/2}, {x: 1/2*sqrt(1205) + 35/2}]

So, you can get each solution by looking at the x values:

sage: [s[x] for s in p.solve(x, solution_dict=True)]
[-1/2*sqrt(1205) + 35/2, 1/2*sqrt(1205) + 35/2]

Then, as before, you can take the maximal element of this list:

sage: max([s[x] for s in p.solve(x, solution_dict=True)])
1/2*sqrt(1205) + 35/2