Solve (k-1)x^2+(3k-1)x-2=0

$$( k - 1 ) x ^ { 2 } + ( 3 k - 1 ) x - 2 = 0$$

Solve for k

$$k=\frac{x^{2}+x+2}{x\left(x+3\right)}$$

$$x\neq -3\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply k-1 by x^{2}.

$$kx^{2}-x^{2}+\left(3k-1\right)x-2=0$$

Use the distributive property to multiply 3k-1 by x.

$$kx^{2}-x^{2}+3kx-x-2=0$$

Add x^{2} to both sides. Anything plus zero gives itself.

$$kx^{2}+3kx-x-2=x^{2}$$

Add x to both sides.

$$kx^{2}+3kx-2=x^{2}+x$$

Add 2 to both sides.

$$kx^{2}+3kx=x^{2}+x+2$$

Combine all terms containing k.

$$\left(x^{2}+3x\right)k=x^{2}+x+2$$

Divide both sides by x^{2}+3x.

$$\frac{\left(x^{2}+3x\right)k}{x^{2}+3x}=\frac{x^{2}+x+2}{x^{2}+3x}$$

Dividing by x^{2}+3x undoes the multiplication by x^{2}+3x.

$$k=\frac{x^{2}+x+2}{x^{2}+3x}$$

Divide x^{2}+x+2 by x^{2}+3x.

$$k=\frac{x^{2}+x+2}{x\left(x+3\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{\left(9k-7\right)\left(k+1\right)}-3k+1}{2\left(k-1\right)}\text{; }x=\frac{-\sqrt{\left(9k-7\right)\left(k+1\right)}-3k+1}{2\left(k-1\right)}\text{, }&k\neq 1\\x=1\text{, }&k=1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{\left(9k-7\right)\left(k+1\right)}-3k+1}{2\left(k-1\right)}\text{; }x=\frac{-\sqrt{\left(9k-7\right)\left(k+1\right)}-3k+1}{2\left(k-1\right)}\text{, }&\left(k\neq 1\text{ and }k\geq \frac{7}{9}\right)\text{ or }k\leq -1\\x=1\text{, }&k=1\end{matrix}\right.$$