Solve x^2+2a^2+2b=+3x(a-b)-4ab

$$x ^ { 2 } + 2 a ^ { 2 } + 2 b = + 3 x ( a - b ) - 4 a b$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{\left(x-a\right)\left(x-2a\right)}{3x+4a+2}\text{, }&x\neq \frac{-4a-2}{3}\\b\in \mathrm{C}\text{, }&\left(x=-\frac{2}{5}\text{ and }a=-\frac{1}{5}\right)\text{ or }\left(x=-\frac{2}{7}\text{ and }a=-\frac{2}{7}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3x by a-b.

$$x^{2}+2a^{2}+2b=3xa-3xb-4ab$$

Add 3xb to both sides.

$$x^{2}+2a^{2}+2b+3xb=3xa-4ab$$

Add 4ab to both sides.

$$x^{2}+2a^{2}+2b+3xb+4ab=3xa$$

Subtract x^{2} from both sides.

$$2a^{2}+2b+3xb+4ab=3xa-x^{2}$$

Subtract 2a^{2} from both sides.

$$2b+3xb+4ab=3xa-x^{2}-2a^{2}$$

Combine all terms containing b.

$$\left(2+3x+4a\right)b=3xa-x^{2}-2a^{2}$$

The equation is in standard form.

$$\left(3x+4a+2\right)b=-x^{2}+3ax-2a^{2}$$

Divide both sides by 2+3x+4a.

$$\frac{\left(3x+4a+2\right)b}{3x+4a+2}=\frac{\left(a-x\right)\left(x-2a\right)}{3x+4a+2}$$

Dividing by 2+3x+4a undoes the multiplication by 2+3x+4a.

$$b=\frac{\left(a-x\right)\left(x-2a\right)}{3x+4a+2}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{\left(x-a\right)\left(x-2a\right)}{3x+4a+2}\text{, }&x\neq \frac{-4a-2}{3}\\b\in \mathrm{R}\text{, }&\left(x=-\frac{2}{5}\text{ and }a=-\frac{1}{5}\right)\text{ or }\left(x=-\frac{2}{7}\text{ and }a=-\frac{2}{7}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3x by a-b.

$$x^{2}+2a^{2}+2b=3xa-3xb-4ab$$

Add 3xb to both sides.

$$x^{2}+2a^{2}+2b+3xb=3xa-4ab$$

Add 4ab to both sides.

$$x^{2}+2a^{2}+2b+3xb+4ab=3xa$$

Subtract x^{2} from both sides.

$$2a^{2}+2b+3xb+4ab=3xa-x^{2}$$

Subtract 2a^{2} from both sides.

$$2b+3xb+4ab=3xa-x^{2}-2a^{2}$$

Combine all terms containing b.

$$\left(2+3x+4a\right)b=3xa-x^{2}-2a^{2}$$

The equation is in standard form.

$$\left(3x+4a+2\right)b=-x^{2}+3ax-2a^{2}$$

Divide both sides by 2+3x+4a.

$$\frac{\left(3x+4a+2\right)b}{3x+4a+2}=\frac{\left(a-x\right)\left(x-2a\right)}{3x+4a+2}$$

Dividing by 2+3x+4a undoes the multiplication by 2+3x+4a.

$$b=\frac{\left(a-x\right)\left(x-2a\right)}{3x+4a+2}$$

Solve for a (complex solution)

$$a=\frac{\sqrt{x^{2}-48bx+16b^{2}-16b}}{4}+\frac{3x}{4}-b$$

$$a=-\frac{\sqrt{x^{2}-48bx+16b^{2}-16b}}{4}+\frac{3x}{4}-b$$

Solve for a

$$a=\frac{\sqrt{x^{2}-48bx+16b^{2}-16b}}{4}+\frac{3x}{4}-b$$

$$a=-\frac{\sqrt{x^{2}-48bx+16b^{2}-16b}}{4}+\frac{3x}{4}-b\text{, }\left(b>-\frac{1}{35}\text{ and }b<0\right)\text{ or }x\geq 4\sqrt{35b^{2}+b}+24b\text{ or }x\leq -4\sqrt{35b^{2}+b}+24b\text{ or }\left(b\geq -\frac{1}{35}\text{ and }b\leq 0\right)$$