Solve (a-b)=a^2-2ab+blambda

$$( a - b ) = a ^ { 2 } - 2 a b + b \lambda$$

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Add 2ab to both sides.

$$a-b+2ab=a^{2}+b\lambda$$

Subtract b\lambda from both sides.

$$a-b+2ab-b\lambda =a^{2}$$

Subtract a from both sides.

$$-b+2ab-b\lambda =a^{2}-a$$

Combine all terms containing b.

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

The equation is in standard form.

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

Divide both sides by -1+2a-\lambda .

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

Dividing by -1+2a-\lambda undoes the multiplication by -1+2a-\lambda .

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

Solve for b

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

Steps for Solving Linear Equation

Add 2ab to both sides.

$$a-b+2ab=a^{2}+b\lambda$$

Subtract b\lambda from both sides.

$$a-b+2ab-b\lambda =a^{2}$$

Subtract a from both sides.

$$-b+2ab-b\lambda =a^{2}-a$$

Combine all terms containing b.

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

The equation is in standard form.

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

Divide both sides by -1+2a-\lambda .

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

Dividing by -1+2a-\lambda undoes the multiplication by -1+2a-\lambda .

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

Solve for a (complex solution)

$$a=-\frac{\sqrt{4b^{2}-4b\lambda +1}}{2}+b+\frac{1}{2}$$

$$a=\frac{\sqrt{4b^{2}-4b\lambda +1}}{2}+b+\frac{1}{2}$$

Solve for a

$$a=-\frac{\sqrt{4b^{2}-4b\lambda +1}}{2}+b+\frac{1}{2}$$

$$a=\frac{\sqrt{4b^{2}-4b\lambda +1}}{2}+b+\frac{1}{2}\text{, }b\geq \frac{\sqrt{\lambda ^{2}-1}+\lambda }{2}\text{ or }b\leq \frac{-\sqrt{\lambda ^{2}-1}+\lambda }{2}\text{ or }|\lambda |\leq 1$$