Solve (a-1)x^2-2ax+a+1=0

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{x+1}{x-1}\text{, }&x\neq 1\\a\in \mathrm{C}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax^{2}-x^{2}-2ax+a+1=0$$

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

$$ax^{2}-2ax+a+1=x^{2}$$

Subtract 1 from both sides.

$$ax^{2}-2ax+a=x^{2}-1$$

Combine all terms containing a.

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

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

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

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

$$a=\frac{x^{2}-1}{x^{2}-2x+1}$$

Divide x^{2}-1 by x^{2}-2x+1.

$$a=\frac{x+1}{x-1}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{x+1}{x-1}\text{, }&x\neq 1\\a\in \mathrm{R}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax^{2}-x^{2}-2ax+a+1=0$$

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

$$ax^{2}-2ax+a+1=x^{2}$$

Subtract 1 from both sides.

$$ax^{2}-2ax+a=x^{2}-1$$

Combine all terms containing a.

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

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

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

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

$$a=\frac{x^{2}-1}{x^{2}-2x+1}$$

Divide x^{2}-1 by x^{2}-2x+1.

$$a=\frac{x+1}{x-1}$$

Solve for x

$$\left\{\begin{matrix}\\x=1\text{, }&\text{unconditionally}\\x=\frac{a+1}{a-1}\text{, }&a\neq 1\end{matrix}\right.$$