Solve x^2+2x=k(x-1)

$$x ^ { 2 } + 2 x = k ( x - 1 )$$

Solve for k

$$k=-\frac{x\left(x+2\right)}{1-x}$$

$$x\neq 1$$

Steps for Solving Linear Equation

Use the distributive property to multiply k by x-1.

$$x^{2}+2x=kx-k$$

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

$$kx-k=x^{2}+2x$$

Combine all terms containing k.

$$\left(x-1\right)k=x^{2}+2x$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)k}{x-1}=\frac{x\left(x+2\right)}{x-1}$$

Dividing by x-1 undoes the multiplication by x-1.

$$k=\frac{x\left(x+2\right)}{x-1}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{k^{2}-8k+4}+k-2}{2}$$

$$x=\frac{-\sqrt{k^{2}-8k+4}+k-2}{2}$$

Solve for x

$$x=\frac{\sqrt{k^{2}-8k+4}+k-2}{2}$$

$$x=\frac{-\sqrt{k^{2}-8k+4}+k-2}{2}\text{, }k\geq 2\sqrt{3}+4\text{ or }k\leq 4-2\sqrt{3}$$