Solve y=Ax+Bsqrt{1+x^2}Rightarrowy=Ax+B(1+x^2)^1/2

$$y = A x + B \sqrt { 1 + x ^ { 2 } } \Rightarrow y = A x + B ( 1 + x ^ { 2 } ) ^ { 1 / 2 }$$

Solve for A

$$\left\{\begin{matrix}A=-\frac{B\sqrt{x^{2}+1}-y}{x}\text{, }&x\neq 0\\A\in \mathrm{R}\text{, }&y=B\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$Ax+B\sqrt{1+x^{2}}=y$$

Subtract B\sqrt{1+x^{2}} from both sides.

$$Ax=y-B\sqrt{1+x^{2}}$$

Reorder the terms.

$$Ax=-B\sqrt{x^{2}+1}+y$$

The equation is in standard form.

$$xA=-B\sqrt{x^{2}+1}+y$$

Divide both sides by x.

$$\frac{xA}{x}=\frac{-B\sqrt{x^{2}+1}+y}{x}$$

Dividing by x undoes the multiplication by x.

$$A=\frac{-B\sqrt{x^{2}+1}+y}{x}$$

Solve for B

$$B=-\frac{Ax-y}{\sqrt{x^{2}+1}}$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$Ax+B\sqrt{1+x^{2}}=y$$

Subtract Ax from both sides.

$$B\sqrt{1+x^{2}}=y-Ax$$

The equation is in standard form.

$$\sqrt{x^{2}+1}B=y-Ax$$

Divide both sides by \sqrt{1+x^{2}}.

$$\frac{\sqrt{x^{2}+1}B}{\sqrt{x^{2}+1}}=\frac{y-Ax}{\sqrt{x^{2}+1}}$$

Dividing by \sqrt{1+x^{2}} undoes the multiplication by \sqrt{1+x^{2}}.

$$B=\frac{y-Ax}{\sqrt{x^{2}+1}}$$