Solve ax^2+3x-a+1=0

$$a x ^ { 2 } + 3 x - a + 1 = 0$$

Solve for a (complex solution)

$$a=-\frac{3x+1}{x^{2}-1}$$

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

Steps for Solving Linear Equation

Subtract 3x from both sides. Anything subtracted from zero gives its negation.

$$ax^{2}-a+1=-3x$$

Subtract 1 from both sides.

$$ax^{2}-a=-3x-1$$

Combine all terms containing a.

$$\left(x^{2}-1\right)a=-3x-1$$

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

$$\frac{\left(x^{2}-1\right)a}{x^{2}-1}=\frac{-3x-1}{x^{2}-1}$$

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

$$a=\frac{-3x-1}{x^{2}-1}$$

Divide -3x-1 by x^{2}-1.

$$a=-\frac{3x+1}{x^{2}-1}$$

Solve for a

$$a=-\frac{3x+1}{x^{2}-1}$$

$$|x|\neq 1$$

Steps for Solving Linear Equation

Subtract 3x from both sides. Anything subtracted from zero gives its negation.

$$ax^{2}-a+1=-3x$$

Subtract 1 from both sides.

$$ax^{2}-a=-3x-1$$

Combine all terms containing a.

$$\left(x^{2}-1\right)a=-3x-1$$

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

$$\frac{\left(x^{2}-1\right)a}{x^{2}-1}=\frac{-3x-1}{x^{2}-1}$$

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

$$a=\frac{-3x-1}{x^{2}-1}$$

Divide -3x-1 by x^{2}-1.

$$a=-\frac{3x+1}{x^{2}-1}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{4a^{2}-4a+9}-3}{2a}\text{; }x=-\frac{\sqrt{4a^{2}-4a+9}+3}{2a}\text{, }&a\neq 0\\x=-\frac{1}{3}\text{, }&a=0\end{matrix}\right.$$