Solve y=2xsqrt{y-1}

$$y = 2 x \sqrt { y - 1 }$$

Solve for x (complex solution)

$$x=\frac{\left(y-1\right)^{-\frac{1}{2}}y}{2}$$

$$y\neq 1$$

Steps for Solving Linear Equation

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

$$2x\sqrt{y-1}=y$$

The equation is in standard form.

$$2\sqrt{y-1}x=y$$

Divide both sides by 2\sqrt{y-1}.

$$\frac{2\sqrt{y-1}x}{2\sqrt{y-1}}=\frac{y}{2\sqrt{y-1}}$$

Dividing by 2\sqrt{y-1} undoes the multiplication by 2\sqrt{y-1}.

$$x=\frac{y}{2\sqrt{y-1}}$$

Divide y by 2\sqrt{y-1}.

$$x=\frac{\left(y-1\right)^{-\frac{1}{2}}y}{2}$$

Solve for x

$$x=\frac{y}{2\sqrt{y-1}}$$

$$y>1$$

Steps for Solving Linear Equation

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

$$2x\sqrt{y-1}=y$$

The equation is in standard form.

$$2\sqrt{y-1}x=y$$

Divide both sides by 2\sqrt{y-1}.

$$\frac{2\sqrt{y-1}x}{2\sqrt{y-1}}=\frac{y}{2\sqrt{y-1}}$$

Dividing by 2\sqrt{y-1} undoes the multiplication by 2\sqrt{y-1}.

$$x=\frac{y}{2\sqrt{y-1}}$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=2\left(\sqrt{x^{2}\left(x^{2}-1\right)}+x^{2}\right)\text{, }&|arg(x\sqrt{2\sqrt{x^{4}-x^{2}}+2x^{2}-1})-arg(2\sqrt{x^{4}-x^{2}}+2x^{2})|<\pi \\y=2\left(-\sqrt{x^{2}\left(x^{2}-1\right)}+x^{2}\right)\text{, }&|arg(x\sqrt{-2\sqrt{x^{4}-x^{2}}+2x^{2}-1})-arg(-2\sqrt{x^{4}-x^{2}}+2x^{2})|<\pi \end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=2x\left(\sqrt{x^{2}-1}+x\right)\text{, }&x\left(\sqrt{x^{2}-1}+x\right)\geq \frac{1}{2}\text{ and }x\sqrt{2x\sqrt{x^{2}-1}+2x^{2}-1}\geq 0\text{ and }|x|\geq 1\\y=2x\left(-\sqrt{x^{2}-1}+x\right)\text{, }&-x\sqrt{x^{2}-1}+x^{2}\geq \frac{1}{2}\text{ and }x\sqrt{-2x\sqrt{x^{2}-1}+2x^{2}-1}\geq 0\text{ and }-2x\sqrt{x^{2}-1}+2x^{2}\geq 1\text{ and }|x|\geq 1\end{matrix}\right.$$