I think that using Newton's method on the polynomial f(x) = x^{2} - B is

the simplest way to find good approximations of the square root of B:

set A_{0} to a guess. If you don't have a guess just let A_{0} = B and

A_{n+1} = ^{(An2 + B)}/_{2An }until you get a good enough approximation.

For example if B is 236 then I know that 15^{2} is 225 so I would let x_{0} = 15. This gives

A_{1} = ^{(152 + 236)}/_{2(15)} = ^{461}/_{30} = 15.3667

15.3667^{2} = 236.1355 so that may be close enough. If not try A_{2}

A_{2} = ^{(15.36672 + 236)}/_{2(15.3667)} = ^{472.1355}/_{2(15.3667)} = 15.3623

15.3623^{2} = 236.0002

I can, for instance take

(x+2)^{2} = x^{2} + 4x + 4

and subtract 4 on both sides to get

(x+2)^{2} - 4 <= x^{2} + 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

same.