Solve (a+b-2c)x^2+(2a-b-c)x+(c+a-2b)=0

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

Solve for a (complex solution)

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

Steps for Solving Linear Equation

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

$$ax^{2}+bx^{2}-2cx^{2}+\left(2a-b-c\right)x+c+a-2b=0$$

Use the distributive property to multiply 2a-b-c by x.

$$ax^{2}+bx^{2}-2cx^{2}+2ax-bx-cx+c+a-2b=0$$

Subtract bx^{2} from both sides. Anything subtracted from zero gives its negation.

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

Add 2cx^{2} to both sides.

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

Add bx to both sides.

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

Add cx to both sides.

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

Subtract c from both sides.

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

Add 2b to both sides.

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

Reorder the terms.

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

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

Divide \left(1+x\right)\left(2b-c-xb+2cx\right) by x^{2}+2x+1.

$$a=\frac{-bx+2cx+2b-c}{x+1}$$

Solve for b (complex solution)

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

Steps for Solving Linear Equation

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

$$ax^{2}+bx^{2}-2cx^{2}+\left(2a-b-c\right)x+c+a-2b=0$$

Use the distributive property to multiply 2a-b-c by x.

$$ax^{2}+bx^{2}-2cx^{2}+2ax-bx-cx+c+a-2b=0$$

Subtract ax^{2} from both sides. Anything subtracted from zero gives its negation.

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

Add 2cx^{2} to both sides.

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

Subtract 2ax from both sides.

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

Add cx to both sides.

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

Subtract c from both sides.

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

Subtract a from both sides.

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

Reorder the terms.

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

Combine all terms containing b.

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

The equation is in standard form.

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

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

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

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

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

Divide \left(1+x\right)\left(-a-c-xa+2cx\right) by x^{2}-x-2.

$$b=\frac{-ax+2cx-a-c}{x-2}$$

Solve for a

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

Steps for Solving Linear Equation

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

$$ax^{2}+bx^{2}-2cx^{2}+\left(2a-b-c\right)x+c+a-2b=0$$

Use the distributive property to multiply 2a-b-c by x.

$$ax^{2}+bx^{2}-2cx^{2}+2ax-bx-cx+c+a-2b=0$$

Subtract bx^{2} from both sides. Anything subtracted from zero gives its negation.

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

Add 2cx^{2} to both sides.

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

Add bx to both sides.

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

Add cx to both sides.

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

Subtract c from both sides.

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

Add 2b to both sides.

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

Reorder the terms.

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

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

Divide \left(1+x\right)\left(2b-c-xb+2cx\right) by x^{2}+2x+1.

$$a=\frac{-bx+2cx+2b-c}{x+1}$$

Solve for b

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

Steps for Solving Linear Equation

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

$$ax^{2}+bx^{2}-2cx^{2}+\left(2a-b-c\right)x+c+a-2b=0$$

Use the distributive property to multiply 2a-b-c by x.

$$ax^{2}+bx^{2}-2cx^{2}+2ax-bx-cx+c+a-2b=0$$

Subtract ax^{2} from both sides. Anything subtracted from zero gives its negation.

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

Add 2cx^{2} to both sides.

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

Subtract 2ax from both sides.

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

Add cx to both sides.

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

Subtract c from both sides.

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

Subtract a from both sides.

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

Reorder the terms.

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

Combine all terms containing b.

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

The equation is in standard form.

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

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

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

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

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

Divide \left(1+x\right)\left(-a-c-xa+2cx\right) by x^{2}-x-2.

$$b=\frac{-ax+2cx-a-c}{x-2}$$