Why is the quadratic equation $ax^2+bx+c=0$? Shouldn't it be $y=ax^2+bx+c$? According to Wikipedia, it is $ax^2+bx+c=0$.
I guess that they are both equations, right?
 A: It seems to me that conventional usage is sloppy, and has been so for a long time. So, it’s perfectly understandable that anyone would be confused.
Let me give an idiosyncratic, stuffy, formalistic answer to your question. As far as I’m concerned, an equation is any mathematical statement in which the verb is an equals sign, “$=$”. Thus both “$ax^2+bx+c=0$” and $y=ax^2+bx+c$” are quadratic equations. What is not a quadratic equation is ”$ax^2+bx+c$”.
When one speaks of “solving” an equation of any kind, one ordinarily means taking the equation and finding all values of the indeterminate(s) that make the sentence true. In this case, solving $y=ax^2+bx+c$ would mean drawing the graph, I suppose, and this is certainly never what one means by the expression “solving the quadratic equation”. On the other hand, $ax^2+bx+c=0$ is an equation that has only two solutions, and that’s the one where the quadratic formula comes in. Others may disagree violently with my position; but as my grandmother would say, You pays your money and takes your choice.
A: They are both equations that involve quadratics, but the usual meaning of "quadratic equation" is that it has only one variable, while $y=ax^2+bx+c$ has two variables ($x$ and $y$).
A: $$ax^2+bx+c=y\iff ax^2+bx+(c-y)=0\iff ax^2+bx+c'=0$$
