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

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{-x^{2}+bx+c-b}{\left(x-2\right)\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 2\\a\in \mathrm{C}\text{, }&\left(b=4-c\text{ and }x=2\right)\text{ or }\left(c=1\text{ and }x=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply ax-a by x-2 and combine like terms.

$$x^{2}=ax^{2}-3ax+2a+b\left(x-1\right)+c$$

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

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

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

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

Subtract bx from both sides.

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

Add b to both sides.

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

Subtract c from both sides.

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

Combine all terms containing a.

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

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

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

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

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

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

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

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{ax^{2}-x^{2}-3ax+c+2a}{x-1}\text{, }&x\neq 1\\b\in \mathrm{C}\text{, }&c=1\text{ and }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply ax-a by x-2 and combine like terms.

$$x^{2}=ax^{2}-3ax+2a+b\left(x-1\right)+c$$

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

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

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

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

Subtract ax^{2} from both sides.

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

Add 3ax to both sides.

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

Subtract 2a from both sides.

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

Subtract c from both sides.

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

Reorder the terms.

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

Combine all terms containing b.

$$\left(x-1\right)b=-ax^{2}+x^{2}+3ax-2a-c$$

The equation is in standard form.

$$\left(x-1\right)b=-ax^{2}+x^{2}+3ax-c-2a$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)b}{x-1}=\frac{-ax^{2}+x^{2}+3ax-c-2a}{x-1}$$

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

$$b=\frac{-ax^{2}+x^{2}+3ax-c-2a}{x-1}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{-x^{2}+bx+c-b}{\left(x-2\right)\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 2\\a\in \mathrm{R}\text{, }&\left(b=4-c\text{ and }x=2\right)\text{ or }\left(c=1\text{ and }x=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply ax-a by x-2 and combine like terms.

$$x^{2}=ax^{2}-3ax+2a+b\left(x-1\right)+c$$

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

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

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

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

Subtract bx from both sides.

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

Add b to both sides.

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

Subtract c from both sides.

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

Combine all terms containing a.

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

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

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

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

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

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

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

Solve for b

$$\left\{\begin{matrix}b=-\frac{ax^{2}-x^{2}-3ax+c+2a}{x-1}\text{, }&x\neq 1\\b\in \mathrm{R}\text{, }&c=1\text{ and }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply ax-a by x-2 and combine like terms.

$$x^{2}=ax^{2}-3ax+2a+b\left(x-1\right)+c$$

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

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

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

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

Subtract ax^{2} from both sides.

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

Add 3ax to both sides.

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

Subtract 2a from both sides.

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

Subtract c from both sides.

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

Reorder the terms.

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

Combine all terms containing b.

$$\left(x-1\right)b=-ax^{2}+x^{2}+3ax-2a-c$$

The equation is in standard form.

$$\left(x-1\right)b=-ax^{2}+x^{2}+3ax-c-2a$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)b}{x-1}=\frac{-ax^{2}+x^{2}+3ax-c-2a}{x-1}$$

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

$$b=\frac{-ax^{2}+x^{2}+3ax-c-2a}{x-1}$$