How do you solve $y=ax^2-ax+b$ for $x$?

Question

I have the polynomial $y = ax^2 - ax + b$, where I can find $y$ given that I know $a$, $b$, and $x$.

How can I transform this polynomial to give me $x$ when I know $a$, $b$, and $y$?

Answer 1

Following the suggestion in the comments, $$x = \frac{a \pm \sqrt{a^2 - 4a(b-y)}}{2a}\,.$$

Answer 2

$$ax^2-ax+(b-y)=0$$

$$x = \frac{a \pm \sqrt{a^2 - 4a(b-y)}}{2a}\,.$$

Only when a≠0