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Try

sage: solve(sqrt(-x^2 + 1)==-4/3*x - 25/36, x, to_poly_solve=True)
[x == -1/20*sqrt(119) - 1/3]

or

sage: solve(sqrt(-x^2 + 1)==-4/3*x - 25/36, x, algorithm='sympy')
[x == -1/20*sqrt(119) - 1/3]