Do two equivalent quadratic forms necessarily have the same solutions

Question

Do two equivalent quadratic forms necessarily have the same solutions? Suppose that I have $Q(x,y)= x^{2}- xy+ 8y^{2}$ and $R(x,y)= 2x^{2}+ 3xy+ 5y^{2}$ and the value of $Q(2,1)$ and $R(2,1$) are different. Does that mean they are not equivalent quadratic forms?

Answer 1

Let us take the example of quadratic forms

$$Q_1(x,y)=x^2-y^2 \ \ \ \text{and} \ \ \ Q_2(x,y)=2xy$$

They are equivalent, due to matrix identity : $A_1 = P^TA_2P$, for a certain matrix $P$, more precisely, with $a:=\frac{1}{\sqrt{2}}$ :

$$\begin{pmatrix}1&0\\0&-1\end{pmatrix}=\begin{pmatrix}a&a\\a&-a\end{pmatrix} \begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}a&a\\a&-a\end{pmatrix},$$

but the zero sets $x^2-y^2=0$ and $2xy=0$ are not at all the same (the two axes' bissectors in the first case and the two axes in the second one).

Answer 2

The quadratic forms $P(x,y)=2x^2+y^2$ and $Q(x,y)=6x^2+16xy+12y^2$ are equivalent. However, $P(2,1)\neq Q(2,1)$.