01 Q ( q ) = k 1 q 1 2 + k 2 q 2 2 + k 3 q 3 2 + k 4 q 4 2 , ∑ i = 1 4 k i = 0 Q(q)=k_1q_1^2+k_2q_2^2+k_3q_3^2+k_4q_4^2,\qquad \sum_{i=1}^4k_i=0 Q ( q ) = k 1 q 1 2 + k 2 q 2 2 + k 3 q 3 2 + k 4 q 4 2 , i = 1 ∑ 4 k i = 0 02 H = ( 1 1 1 1 1 1 − 1 − 1 1 − 1 1 − 1 1 − 1 − 1 1 ) , det H = − 16 H=\begin{pmatrix}1&1&1&1\\1&1&-1&-1\\1&-1&1&-1\\1&-1&-1&1\end{pmatrix},\qquad \det H=-16 H = 1 1 1 1 1 1 − 1 − 1 1 − 1 1 − 1 1 − 1 − 1 1 , det H = − 16 03 ε ∈ rows ( H ) , D = ∑ k i u i 2 , L ε = ∑ k i ε i u i , q i = D ε i − 2 L ε u i \varepsilon\in\operatorname{rows}(H),\quad D=\sum k_iu_i^2,\quad L_\varepsilon=\sum k_i\varepsilon_i u_i,\quad q_i=D\varepsilon_i-2L_\varepsilon u_i ε ∈ rows ( H ) , D = ∑ k i u i 2 , L ε = ∑ k i ε i u i , q i = D ε i − 2 L ε u i 04 Q ( q ) = D 2 Q ( ε ) − 4 D L ε 2 + 4 L ε 2 D = 0 Q(q)=D^2Q(\varepsilon)-4DL_\varepsilon^2+4L_\varepsilon^2D=0 Q ( q ) = D 2 Q ( ε ) − 4 D L ε 2 + 4 L ε 2 D = 0 05 Q ( q ∗ ) = 0 : H ( k 1 q 1 ∗ , … , k 4 q 4 ∗ ) T ≠ 0 ⟹ ∃ ε : L ε ( q ∗ ) ≠ 0 Q(q^*)=0:\quad H(k_1q_1^*,\ldots,k_4q_4^*)^T\ne0\Longrightarrow\exists\varepsilon:\ L_\varepsilon(q^*)\ne0 Q ( q ∗ ) = 0 : H ( k 1 q 1 ∗ , … , k 4 q 4 ∗ ) T = 0 ⟹ ∃ ε : L ε ( q ∗ ) = 0 06 u = q ∗ ⟹ D = 0 , q = − 2 L ε ( q ∗ ) q ∗ u=q^*\Longrightarrow D=0,\qquad q=-2L_\varepsilon(q^*)q^* u = q ∗ ⟹ D = 0 , q = − 2 L ε ( q ∗ ) q ∗ The four rows of H cover every nonzero rational point when kᵢ ≠ 0. A common denominator is cleared by homogeneous root scaling; a zero coefficient is handled separately as the red conic with one free root.