Solve (k-1)x^2-6x+(k-1)=0

$$( k - 1 ) x ^ { 2 } - 6 x + ( k - 1 ) = 0$$

Solve for k (complex solution)

$$k=\frac{x^{2}+6x+1}{x^{2}+1}$$

$$x\neq -i\text{ and }x\neq i$$

Steps for Solving Linear Equation

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

$$kx^{2}-x^{2}-6x+k-1=0$$

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

$$kx^{2}-6x+k-1=x^{2}$$

Add 6x to both sides.

$$kx^{2}+k-1=x^{2}+6x$$

Add 1 to both sides.

$$kx^{2}+k=x^{2}+6x+1$$

Combine all terms containing k.

$$\left(x^{2}+1\right)k=x^{2}+6x+1$$

Divide both sides by x^{2}+1.

$$\frac{\left(x^{2}+1\right)k}{x^{2}+1}=\frac{x^{2}+6x+1}{x^{2}+1}$$

Dividing by x^{2}+1 undoes the multiplication by x^{2}+1.

$$k=\frac{x^{2}+6x+1}{x^{2}+1}$$

Solve for k

$$k=\frac{x^{2}+6x+1}{x^{2}+1}$$

Steps for Solving Linear Equation

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

$$kx^{2}-x^{2}-6x+k-1=0$$

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

$$kx^{2}-6x+k-1=x^{2}$$

Add 6x to both sides.

$$kx^{2}+k-1=x^{2}+6x$$

Add 1 to both sides.

$$kx^{2}+k=x^{2}+6x+1$$

Combine all terms containing k.

$$\left(x^{2}+1\right)k=x^{2}+6x+1$$

Divide both sides by x^{2}+1.

$$\frac{\left(x^{2}+1\right)k}{x^{2}+1}=\frac{x^{2}+6x+1}{x^{2}+1}$$

Dividing by x^{2}+1 undoes the multiplication by x^{2}+1.

$$k=\frac{x^{2}+6x+1}{x^{2}+1}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{-\left(k-4\right)\left(k+2\right)}+3}{k-1}\text{; }x=\frac{-\sqrt{-\left(k-4\right)\left(k+2\right)}+3}{k-1}\text{, }&k\neq 1\\x=0\text{, }&k=1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{\left(4-k\right)\left(k+2\right)}+3}{k-1}\text{; }x=\frac{-\sqrt{\left(4-k\right)\left(k+2\right)}+3}{k-1}\text{, }&k\neq 1\text{ and }k\leq 4\text{ and }k\geq -2\\x=0\text{, }&k=1\end{matrix}\right.$$