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

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{c}{\left(x-2\right)\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 2\\a\in \mathrm{C}\text{, }&\left(x=2\text{ or }x=1\right)\text{ and }c=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

$$ax^{2}-2ax+a+c=ax-a$$

Subtract ax from both sides.

$$ax^{2}-2ax+a+c-ax=-a$$

Combine -2ax and -ax to get -3ax.

$$ax^{2}-3ax+a+c=-a$$

Add a to both sides.

$$ax^{2}-3ax+a+c+a=0$$

Combine a and a to get 2a.

$$ax^{2}-3ax+2a+c=0$$

Subtract c from both sides. Anything subtracted from zero gives its negation.

$$ax^{2}-3ax+2a=-c$$

Combine all terms containing a.

$$\left(x^{2}-3x+2\right)a=-c$$

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

$$\frac{\left(x^{2}-3x+2\right)a}{x^{2}-3x+2}=-\frac{c}{x^{2}-3x+2}$$

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

$$a=-\frac{c}{x^{2}-3x+2}$$

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

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

Solve for a

$$\left\{\begin{matrix}a=-\frac{c}{\left(x-2\right)\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 2\\a\in \mathrm{R}\text{, }&\left(x=2\text{ or }x=1\right)\text{ and }c=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

$$ax^{2}-2ax+a+c=ax-a$$

Subtract ax from both sides.

$$ax^{2}-2ax+a+c-ax=-a$$

Combine -2ax and -ax to get -3ax.

$$ax^{2}-3ax+a+c=-a$$

Add a to both sides.

$$ax^{2}-3ax+a+c+a=0$$

Combine a and a to get 2a.

$$ax^{2}-3ax+2a+c=0$$

Subtract c from both sides. Anything subtracted from zero gives its negation.

$$ax^{2}-3ax+2a=-c$$

Combine all terms containing a.

$$\left(x^{2}-3x+2\right)a=-c$$

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

$$\frac{\left(x^{2}-3x+2\right)a}{x^{2}-3x+2}=-\frac{c}{x^{2}-3x+2}$$

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

$$a=-\frac{c}{x^{2}-3x+2}$$

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

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

Solve for c

$$c=-a\left(x-2\right)\left(x-1\right)$$