Solve ax^2=a+b-bx

$$a x ^ { 2 } = a + b - b x$$

Solve for a (complex solution)

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

Steps for Solving Linear Equation

Subtract a from both sides.

$$ax^{2}-a=b-bx$$

Combine all terms containing a.

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

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

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

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

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

Divide b-bx by x^{2}-1.

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

Solve for b (complex solution)

$$\left\{\begin{matrix}\\b=-a\left(x+1\right)\text{, }&\text{unconditionally}\\b\in \mathrm{C}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Subtract a from both sides.

$$b-bx=ax^{2}-a$$

Combine all terms containing b.

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

Divide both sides by 1-x.

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

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

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

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

$$b=-a\left(x+1\right)$$

Solve for a

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

Steps for Solving Linear Equation

Subtract a from both sides.

$$ax^{2}-a=b-bx$$

Combine all terms containing a.

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

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

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

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

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

Divide b-bx by x^{2}-1.

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

Solve for b

$$\left\{\begin{matrix}\\b=-a\left(x+1\right)\text{, }&\text{unconditionally}\\b\in \mathrm{R}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Subtract a from both sides.

$$b-bx=ax^{2}-a$$

Combine all terms containing b.

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

Divide both sides by 1-x.

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

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

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

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

$$b=-a\left(x+1\right)$$