How can $\sqrt{x + b}$ be transformed into $a\sqrt{x} + c$?

Question

How can a square root function containing a variable and a constant be reexpressed as a square root function containing only a variable?

Answer 1

It cannot, since $\sqrt{x+b}$ is defined for $x=-b$, but $a\sqrt x+c$ is undefined at $x=-b$.

Now, you can approximate as follows:

$$\sqrt{x+b}\approx\sqrt x+\frac b{2\sqrt x}$$

which is the two term binomial expansion for large $x$. For small $x$, to avoid division by $0$, one should invert it:

$$\sqrt{x+b}\approx\sqrt b+\frac x{2\sqrt b}$$