Solve x=frac{sqrt{a+bx}}{t}

$$x = \frac { \sqrt { a + b x } } { t }$$

Solve for a

$$a=x\left(xt^{2}-b\right)$$

$$\left(x\leq 0\text{ and }t<0\right)\text{ or }\left(x\geq 0\text{ and }t>0\right)$$

Steps for Solving Radical Equations

Multiply both sides of the equation by t.

$$xt=\sqrt{a+bx}$$

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

$$\sqrt{a+bx}=xt$$

Square both sides of the equation.

$$a+bx=t^{2}x^{2}$$

Subtract bx from both sides of the equation.

$$a+bx-bx=t^{2}x^{2}-bx$$

Subtracting bx from itself leaves 0.

$$a=t^{2}x^{2}-bx$$

Subtract bx from x^{2}t^{2}.

$$a=x\left(xt^{2}-b\right)$$

Solve for b

$$\left\{\begin{matrix}b=\frac{\left(tx\right)^{2}-a}{x}\text{, }&\left(x<0\text{ and }t<0\right)\text{ or }\left(x>0\text{ and }t>0\right)\\b\in \mathrm{R}\text{, }&x=0\text{ and }a=0\text{ and }t\neq 0\end{matrix}\right.$$

Steps for Solving Radical Equations

Multiply both sides of the equation by t.

$$xt=\sqrt{a+bx}$$

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

$$\sqrt{a+bx}=xt$$

Square both sides of the equation.

$$xb+a=t^{2}x^{2}$$

Subtract a from both sides of the equation.

$$xb+a-a=t^{2}x^{2}-a$$

Subtracting a from itself leaves 0.

$$xb=t^{2}x^{2}-a$$

Divide both sides by x.

$$\frac{xb}{x}=\frac{t^{2}x^{2}-a}{x}$$

Dividing by x undoes the multiplication by x.

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

Divide x^{2}t^{2}-a by x.

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

Solve for a (complex solution)

$$a=x\left(xt^{2}-b\right)$$

$$\left(x=0\text{ or }arg(tx)<\pi \right)\text{ and }t\neq 0$$

Solve for b (complex solution)

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