Let $K$ be a field of characteristic $2$ ($2$ is very important in the statement -- otherwise I can do it myself :) ). On the set $K \times K$ we define the following equivalent relation: $(a, b)  \equiv (a', b') $ if and only if  there exists a pair $(q, \alpha) \in K^* \times K$ such
that:
$$
a = q^2  a' + \alpha^2 - b  \alpha \quad {\rm and} \quad 
b = q b' 
$$

Can you compute explicitely the quotient set $K\times K/\equiv $? Or, if you prefere, can you give a set of representatives for the relation $\equiv$.