Solve y=ax^2-(a-2)x+1

$$y = a x ^ { 2 } - ( a - 2 ) x + 1$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{2x-y+1}{x\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 0\\a\in \mathrm{C}\text{, }&\left(y=1\text{ and }x=0\right)\text{ or }\left(y=3\text{ and }x=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

To find the opposite of ax-2x, find the opposite of each term.

$$y=ax^{2}-ax+2x+1$$

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

$$ax^{2}-ax+2x+1=y$$

Subtract 2x from both sides.

$$ax^{2}-ax+1=y-2x$$

Subtract 1 from both sides.

$$ax^{2}-ax=y-2x-1$$

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

Divide y-2x-1 by x^{2}-x.

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

Solve for a

$$\left\{\begin{matrix}a=-\frac{2x-y+1}{x\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&\left(y=1\text{ and }x=0\right)\text{ or }\left(y=3\text{ and }x=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

To find the opposite of ax-2x, find the opposite of each term.

$$y=ax^{2}-ax+2x+1$$

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

$$ax^{2}-ax+2x+1=y$$

Subtract 2x from both sides.

$$ax^{2}-ax+1=y-2x$$

Subtract 1 from both sides.

$$ax^{2}-ax=y-2x-1$$

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

Divide y-2x-1 by x^{2}-x.

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{4ay+a^{2}-8a+4}+a-2}{2a}\text{; }x=\frac{-\sqrt{4ay+a^{2}-8a+4}+a-2}{2a}\text{, }&a\neq 0\\x=\frac{y-1}{2}\text{, }&a=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{4ay+a^{2}-8a+4}+a-2}{2a}\text{; }x=\frac{-\sqrt{4ay+a^{2}-8a+4}+a-2}{2a}\text{, }&\left(a>0\text{ or }y\leq -\frac{a^{2}-8a+4}{4a}\right)\text{ and }\left(y\leq \text{Indeterminate}\text{ or }a\neq 0\right)\text{ and }\left(a<0\text{ or }\left(a\neq 0\text{ and }y\geq -\frac{a^{2}-8a+4}{4a}\right)\right)\\x=\frac{y-1}{2}\text{, }&a=0\end{matrix}\right.$$