When are two quadratic forms equivalent?

$Q_1$. $Q_1=(x+2y)^2-y^2$, signature $(1,1,0)$.

$Q_2$. $Q_2=x^2-y^2$, signature $(1,1,0)$.

Equivalence. Two quadratic forms are equivalent iff they have the same signature (Sylvester's law of inertia). Both have signature $(1,1,0)$, so they are equivalent.

Explicit change. $Q_1=(x+2y)^2-y^2$. Set $u=x+2y$, $v=y$. Then $Q_1=u^2-v^2$, which is $Q_2$ in the variables $(u,v)$.

Conclusion. $Q_1$ and $Q_2$ are equivalent.