Solve y=axsqrt{b}

$$y = a x \sqrt { b }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{b^{-\frac{1}{2}}y}{x}\text{, }&b\neq 0\text{ and }x\neq 0\\a\in \mathrm{C}\text{, }&\left(b=0\text{ or }x=0\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax\sqrt{b}=y$$

The equation is in standard form.

$$\sqrt{b}xa=y$$

Divide both sides by x\sqrt{b}.

$$\frac{\sqrt{b}xa}{\sqrt{b}x}=\frac{y}{\sqrt{b}x}$$

Dividing by x\sqrt{b} undoes the multiplication by x\sqrt{b}.

$$a=\frac{y}{\sqrt{b}x}$$

Divide y by x\sqrt{b}.

$$a=\frac{b^{-\frac{1}{2}}y}{x}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{y}{\sqrt{b}x}\text{, }&x\neq 0\text{ and }b>0\\a\in \mathrm{R}\text{, }&\left(y=0\text{ and }x=0\text{ and }b>0\right)\text{ or }\left(b=0\text{ and }y=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax\sqrt{b}=y$$

The equation is in standard form.

$$\sqrt{b}xa=y$$

Divide both sides by x\sqrt{b}.

$$\frac{\sqrt{b}xa}{\sqrt{b}x}=\frac{y}{\sqrt{b}x}$$

Dividing by x\sqrt{b} undoes the multiplication by x\sqrt{b}.

$$a=\frac{y}{\sqrt{b}x}$$

Solve for b

$$\left\{\begin{matrix}b=\left(\frac{y}{ax}\right)^{2}\text{, }&x\neq 0\text{ and }a\neq 0\text{ and }\left(a<0\text{ or }x<0\text{ or }y\geq 0\right)\text{ and }\left(x<0\text{ or }a>0\text{ or }y\leq 0\right)\text{ and }\left(a<0\text{ or }x>0\text{ or }y\leq 0\right)\text{ and }\left(x>0\text{ or }a>0\text{ or }y\geq 0\right)\\b\geq 0\text{, }&\left(y=0\text{ and }x=0\right)\text{ or }\left(x\neq 0\text{ and }y=0\text{ and }a=0\right)\end{matrix}\right.$$

Steps for Solving Radical Equations

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

$$ax\sqrt{b}=y$$

Divide both sides by ax.

$$\frac{ax\sqrt{b}}{ax}=\frac{y}{ax}$$

Dividing by ax undoes the multiplication by ax.

$$\sqrt{b}=\frac{y}{ax}$$

Square both sides of the equation.

$$b=\frac{y^{2}}{\left(ax\right)^{2}}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\left(\frac{y}{ax}\right)^{2}\text{, }&x\neq 0\text{ and }a\neq 0\text{ and }\left(|-arg(y)+arg(\sqrt{\left(\frac{y}{ax}\right)^{2}}ax)|<\pi \text{ or }y=0\right)\\b\in \mathrm{C}\text{, }&\left(x=0\text{ or }a=0\right)\text{ and }y=0\end{matrix}\right.$$