Solve ax^2+(a+b)x+b=0

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

Solve for a (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply a+b by x.

$$ax^{2}+ax+bx+b=0$$

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

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

Subtract b from both sides.

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

Combine all terms containing a.

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

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

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

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

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

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

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

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply a+b by x.

$$ax^{2}+ax+bx+b=0$$

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

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

Subtract ax from both sides.

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

Combine all terms containing b.

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

Divide both sides by 1+x.

$$\frac{\left(x+1\right)b}{x+1}=-\frac{ax\left(x+1\right)}{x+1}$$

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

$$b=-\frac{ax\left(x+1\right)}{x+1}$$

Divide -ax\left(1+x\right) by 1+x.

$$b=-ax$$

Solve for a

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

Steps for Solving Linear Equation

Use the distributive property to multiply a+b by x.

$$ax^{2}+ax+bx+b=0$$

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

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

Subtract b from both sides.

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

Combine all terms containing a.

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

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

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

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

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

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

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

Solve for b

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

Steps for Solving Linear Equation

Use the distributive property to multiply a+b by x.

$$ax^{2}+ax+bx+b=0$$

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

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

Subtract ax from both sides.

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

Combine all terms containing b.

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

Divide both sides by 1+x.

$$\frac{\left(x+1\right)b}{x+1}=-\frac{ax\left(x+1\right)}{x+1}$$

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

$$b=-\frac{ax\left(x+1\right)}{x+1}$$

Divide -ax\left(1+x\right) by 1+x.

$$b=-ax$$