A quadratic form Q(z) is said to be positive definite if Q(z)>0 for z!=0. A real quadratic form in n variables is positive definite iff its canonical form is Q(z) = z_1^2 + z_2^2 + ... + z_n^2. A binary quadratic form F(x, y) = a x^2 + b x y + c y^2 of two real variables is positive definite if it is >0 for any (x, y)!=(0, 0), therefore if a>0 and the binary quadratic form discriminant d congruent 4a c - b^2>0.
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