Quadratic equations in two or more variables

Question

Which of the following is a quadratic equation and why?

$$yx^2+x-1=0$$ $$x^2+x-y=0$$

where $x$ and $y$ are variables. Any help is appreciated.

Answer 1

Rearrange them to put $y$ as the subject and you'll quickly see which is quadratic. A quadratic has the form $y=ax^2+bx+c$

For the first, we get: $$yx^2+x-1=0\to yx^2=1-x\to y=\frac{1}{x^2}-\frac{1}{x}$$ Certainly not a quadratic.

For the second we get: $$x^2+x-y=0\to x^2+x=y\to y=x^2+x$$

This is a quadratic, with $a=1,b=1,c=0$.

Answer 2

First one is a cubic equation in two variables $x$ and $y$ as the first term has degree $3$ the second one is quadratic equation in two variables.