What does it mean to say that a positive definite binary quadratic form is reduced? What does it mean to say that two binary quadratic forms are equivalent? Show that every positive definite binary quadratic form is equivalent to some reduced form.

Show that the reduced positive definite binary quadratic forms of discriminant $-35$ are $f_{1}=x^{2}+x y+9 y^{2}$ and $f_{2}=3 x^{2}+x y+3 y^{2}$. Show also that a prime $p>7$ is represented by $f_{i}$ if and only if

$$\left(\frac{p}{5}\right)=\left(\frac{p}{7}\right)= \begin{cases}+1 & (i=1) \\ -1 & (i=2)\end{cases}$$