I think that using Newton's method on the polynomial f(x) = x2 - B is
the simplest way to find good approximations of the square root of B:
set A0 to a guess. If you don't have a guess just let A0 = B and
An+1 = (An2 + B)/2An until you get a good enough approximation.
For example if B is 236 then I know that 152 is 225 so I would let x0 = 15. This gives
A1 = (152 + 236)/2(15) = 461/30 = 15.3667
15.36672 = 236.1355 so that may be close enough. If not try A2
A2 = (15.36672 + 236)/2(15.3667) = 472.1355/2(15.3667) = 15.3623
15.36232 = 236.0002
I can, for instance take
(x+2)2 = x2 + 4x + 4
and subtract 4 on both sides to get
(x+2)2 - 4 <= x2 + 4x.
However I could not get a strict inequality. A "mathematical operation"
outputs unambiguous results that only depend on the input. The inputs
from both sides of an equation are the same, so the outputs are also the