In general, any equation is a statement of equality between two expressions. In the 2D case, equations generally involve the two variables ##x## and ##y## or either variable alone if we require the other variable to be equal to zero.

The most general quadratic equation should be the polynomial $$ay^2+by+cx^2+dx+e=0$$ where both variables are raised to the 2nd power. Why instead, is the polynomial of the form ##y=ax^2+bx+c## presented in introductory textbooks as the stereotypical and general quadratic equation?

If we set ##y=0##, we get what is called the "quadratic equation": $$ax^2+bx+c=0$$This quadratic equation has only one variable, ##x##, raised to the 2nd power and is solvable using the quadratic formula to find the two solutions, called roots, ##(x_1,0)## and ##(x_2,0)##, which are essentially the -intercepts.

So when I hear that only linear, quadratic, cubic and quartic equations have methods to find their solutions (there are no methods for quintic equations), does it means that there are methods to find the roots of linear, quadratic, cubic and quartic equations involving only the ##x## variable? For example, the solutions of ##0=ax^3+bx^2+cx+d## but not the solutions of equation ##y=ax^3+bx^2+cx+d##.

Thanks.