Solve (1+A)x^2-2x-(1+A)=0

$$( 1 + A ) x ^ { 2 } - 2 x - ( 1 + A ) = 0$$

Solve for A (complex solution)

$$A=\frac{1+2x-x^{2}}{x^{2}-1}$$

$$x\neq -1\text{ and }x\neq 1$$

Steps for Solving Linear Equation

Use the distributive property to multiply 1+A by x^{2}.

$$x^{2}+Ax^{2}-2x-\left(1+A\right)=0$$

To find the opposite of 1+A, find the opposite of each term.

$$x^{2}+Ax^{2}-2x-1-A=0$$

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

$$Ax^{2}-2x-1-A=-x^{2}$$

Add 2x to both sides.

$$Ax^{2}-1-A=-x^{2}+2x$$

Add 1 to both sides.

$$Ax^{2}-A=-x^{2}+2x+1$$

Combine all terms containing A.

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

The equation is in standard form.

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

Divide both sides by x^{2}-1.

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

Dividing by x^{2}-1 undoes the multiplication by x^{2}-1.

$$A=\frac{1+2x-x^{2}}{x^{2}-1}$$

Solve for A

$$A=-\frac{x^{2}-2x-1}{x^{2}-1}$$

$$|x|\neq 1$$

Steps for Solving Linear Equation

Use the distributive property to multiply 1+A by x^{2}.

$$x^{2}+Ax^{2}-2x-\left(1+A\right)=0$$

To find the opposite of 1+A, find the opposite of each term.

$$x^{2}+Ax^{2}-2x-1-A=0$$

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

$$Ax^{2}-2x-1-A=-x^{2}$$

Add 2x to both sides.

$$Ax^{2}-1-A=-x^{2}+2x$$

Add 1 to both sides.

$$Ax^{2}-A=-x^{2}+2x+1$$

Combine all terms containing A.

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

The equation is in standard form.

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

Divide both sides by x^{2}-1.

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

Dividing by x^{2}-1 undoes the multiplication by x^{2}-1.

$$A=\frac{1+2x-x^{2}}{x^{2}-1}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{A^{2}+2A+2}+1}{A+1}\text{; }x=\frac{-\sqrt{A^{2}+2A+2}+1}{A+1}\text{, }&A\neq -1\\x=0\text{, }&A=-1\end{matrix}\right.$$