Solve x^2+2xleft(a+1right)=left(x-aright)^2+2left(x+aright)+1

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

Solve for x

$$x=\frac{\left(a+1\right)^{2}}{4a}$$

$$a\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x by a+1.

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

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

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

Use the distributive property to multiply 2 by x+a.

$$x^{2}+2xa+2x=x^{2}-2xa+a^{2}+2x+2a+1$$

Subtract x^{2} from both sides.

$$x^{2}+2xa+2x-x^{2}=-2xa+a^{2}+2x+2a+1$$

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

$$2xa+2x=-2xa+a^{2}+2x+2a+1$$

Add 2xa to both sides.

$$2xa+2x+2xa=a^{2}+2x+2a+1$$

Combine 2xa and 2xa to get 4xa.

$$4xa+2x=a^{2}+2x+2a+1$$

Subtract 2x from both sides.

$$4xa+2x-2x=a^{2}+2a+1$$

Combine 2x and -2x to get 0.

$$4xa=a^{2}+2a+1$$

The equation is in standard form.

$$4ax=a^{2}+2a+1$$

Divide both sides by 4a.

$$\frac{4ax}{4a}=\frac{\left(a+1\right)^{2}}{4a}$$

Dividing by 4a undoes the multiplication by 4a.

$$x=\frac{\left(a+1\right)^{2}}{4a}$$

Solve for a (complex solution)

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

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

Solve for a

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

$$a=2\sqrt{x\left(x-1\right)}+2x-1\text{, }x\geq 1\text{ or }x\leq 0$$