Solve ax^2+2ax+2bx+b=0

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

Solve for a (complex solution)

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

Steps for Solving Linear Equation

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

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

Subtract b from both sides.

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

Combine all terms containing a.

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

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

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

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

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

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

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

Solve for b (complex solution)

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

Steps for Solving Linear Equation

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

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

Subtract 2ax from both sides.

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

Combine all terms containing b.

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

Divide both sides by 2x+1.

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

Dividing by 2x+1 undoes the multiplication by 2x+1.

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

Solve for a

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

Steps for Solving Linear Equation

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

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

Subtract b from both sides.

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

Combine all terms containing a.

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

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

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

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

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

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

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

Solve for b

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

Steps for Solving Linear Equation

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

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

Subtract 2ax from both sides.

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

Combine all terms containing b.

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

Divide both sides by 2x+1.

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

Dividing by 2x+1 undoes the multiplication by 2x+1.

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