SAT Quadratic Expression

Question

This problem screwed me over on the math section of the SAT. How many of the following are not quadratic forms over $\mathbb R^2$?

I. $x_1x_2$

II. $3x_1^2+7x_1x_2-6x_2^2$

III. $\begin{bmatrix} x&y \end{bmatrix} \begin{bmatrix} 2&3\\3&7 \end{bmatrix}\begin{bmatrix} x\\y \end{bmatrix}$

IV. $\begin{bmatrix} x&y \end{bmatrix} \begin{bmatrix} 2&2\\4&7 \end{bmatrix}\begin{bmatrix} x\\y \end{bmatrix}$

V. $\begin{bmatrix} x&y&1 \end{bmatrix} \begin{bmatrix} 7&-17&7\\-17&17&-17\\7&-17&27 \end{bmatrix}\begin{bmatrix} x\\y\\1 \end{bmatrix}$.

Well, the first reason I didn't get this is because I have no clue what a quadratic form is.

But my intuition tells me that they're polynomials that have degree $2$, hence quadratic. But that means that circles and ovals and hyperbolas are quadratic forms too, but whatever I'll just go with it.

So I can see that $II$, $III$, and $IV$ all satisfy these properties. I don't think $V$ satisfies them because even though there are only $2$ variables, there are some terms that are not quadratic. Also $IV$ and $V$ have some weird symmetry along the diagonals (idk how to describe, idk if it's important either).

And I don't know about $I$, because even though it has only quadratic terms, it doesn't have any single variable squared. But I think it's fine because it's a hyperbola.

So I think the answer is either $1$ or $2$, depending on whether $III$ is one because it doesn't have that weird diagonal thing. And I realized I could rewrite $II$ to be the product of vectors and matrices by putting $3.5$ twice along the diagonal to make it have the diagonal symmetry like $IV$ and $V$.

Answer 1

According wikipedia:

Quadratic forms are homogeneous quadratic polynomials in $n$ variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form:

$$ q(x) = ax^2\quad \textrm{(unary)} \\ q(x,y) = ax^2 + bxy + > cy^2\quad \textrm{(binary)} \\ q(x,y,z) = ax^2 + by^2 + cz^2 + dxy + > exz + fyz \quad\textrm{(ternary)} $$

where $a, ..., f$ are the coefficients.

Note that quadratic functions, such as $ax^2 + bx + c$ in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless b and c are both 0).

Hence all of them are quadratic forms except the last one alternative, which contains constant and linear terms.