Solve x+a/3=(x-b)^2/3x-a+3ab-3b^2/9x-3a

$$\frac { x + a } { 3 } = \frac { ( x - b ) ^ { 2 } } { 3 x - a } + \frac { 3 a b - 3 b ^ { 2 } } { 9 x - 3 a }$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{a}{3}\text{, }&x\neq \frac{a}{3}\\b\in \mathrm{C}\text{, }&x=\frac{a}{2}\text{ and }a\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 3\left(-3x+a\right), the least common multiple of 3,3x-a,9x-3a.

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

Use the distributive property to multiply -3x+a by x+a and combine like terms.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-b\right)^{2}.

$$-3x^{2}-2xa+a^{2}=-3\left(x^{2}-2xb+b^{2}\right)-\left(3ab-3b^{2}\right)$$

Use the distributive property to multiply -3 by x^{2}-2xb+b^{2}.

$$-3x^{2}-2xa+a^{2}=-3x^{2}+6xb-3b^{2}-\left(3ab-3b^{2}\right)$$

To find the opposite of 3ab-3b^{2}, find the opposite of each term.

$$-3x^{2}-2xa+a^{2}=-3x^{2}+6xb-3b^{2}-3ab+3b^{2}$$

Combine -3b^{2} and 3b^{2} to get 0.

$$-3x^{2}-2xa+a^{2}=-3x^{2}+6xb-3ab$$

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

$$-3x^{2}+6xb-3ab=-3x^{2}-2xa+a^{2}$$

Add 3x^{2} to both sides.

$$6xb-3ab=-3x^{2}-2xa+a^{2}+3x^{2}$$

Combine -3x^{2} and 3x^{2} to get 0.

$$6xb-3ab=-2xa+a^{2}$$

Combine all terms containing b.

$$\left(6x-3a\right)b=-2xa+a^{2}$$

The equation is in standard form.

$$\left(6x-3a\right)b=a^{2}-2ax$$

Divide both sides by 6x-3a.

$$\frac{\left(6x-3a\right)b}{6x-3a}=\frac{a\left(a-2x\right)}{6x-3a}$$

Dividing by 6x-3a undoes the multiplication by 6x-3a.

$$b=\frac{a\left(a-2x\right)}{6x-3a}$$

Divide a\left(-2x+a\right) by 6x-3a.

$$b=-\frac{a}{3}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{a}{3}\text{, }&x\neq \frac{a}{3}\\b\in \mathrm{R}\text{, }&x=\frac{a}{2}\text{ and }a\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 3\left(-3x+a\right), the least common multiple of 3,3x-a,9x-3a.

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

Use the distributive property to multiply -3x+a by x+a and combine like terms.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-b\right)^{2}.

$$-3x^{2}-2xa+a^{2}=-3\left(x^{2}-2xb+b^{2}\right)-\left(3ab-3b^{2}\right)$$

Use the distributive property to multiply -3 by x^{2}-2xb+b^{2}.

$$-3x^{2}-2xa+a^{2}=-3x^{2}+6xb-3b^{2}-\left(3ab-3b^{2}\right)$$

To find the opposite of 3ab-3b^{2}, find the opposite of each term.

$$-3x^{2}-2xa+a^{2}=-3x^{2}+6xb-3b^{2}-3ab+3b^{2}$$

Combine -3b^{2} and 3b^{2} to get 0.

$$-3x^{2}-2xa+a^{2}=-3x^{2}+6xb-3ab$$

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

$$-3x^{2}+6xb-3ab=-3x^{2}-2xa+a^{2}$$

Add 3x^{2} to both sides.

$$6xb-3ab=-3x^{2}-2xa+a^{2}+3x^{2}$$

Combine -3x^{2} and 3x^{2} to get 0.

$$6xb-3ab=-2xa+a^{2}$$

Combine all terms containing b.

$$\left(6x-3a\right)b=-2xa+a^{2}$$

The equation is in standard form.

$$\left(6x-3a\right)b=a^{2}-2ax$$

Divide both sides by 6x-3a.

$$\frac{\left(6x-3a\right)b}{6x-3a}=\frac{a\left(a-2x\right)}{6x-3a}$$

Dividing by 6x-3a undoes the multiplication by 6x-3a.

$$b=\frac{a\left(a-2x\right)}{6x-3a}$$

Divide a\left(-2x+a\right) by 6x-3a.

$$b=-\frac{a}{3}$$

Solve for a

$$\left\{\begin{matrix}a=2x\text{, }&x\neq 0\\a=-3b\text{, }&x\neq -b\end{matrix}\right.$$