How do I convert the 3-variable quadratic form q(x₁ x₂ x₃) = row vector [x₁ x₂ x₃] multiplied by [a₁₁ a₁₂ a₁₃ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃]₃ₓ₃ matrix multiplied by column vector [x₁ x₂ x₃] into the equation: $$ |A_1|\left(x_1 + x_2 a_{12}/a_{11} + x_3 a_{13}/a_{11}\right)^2 + (|A₂|/|A₁|)\left(x₂ + x₃(a₁₁a₂₃ - a₁₂a₁₃)/(a₁₁a₁₂ - a²₁₂)\right))^2 + (|A₃|/|A₂|)x^2_3 $$ by "completing the square" method? Please be as detailed as possible, it's doing my head in... thank you.
$$ \left[ \begin{array}{ccc} x_1 & x_2 & x_3 \end{array} \right] \left[ \begin{array}{ccc} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \\ \end{array} \right] \left[ \begin{array}{ccc} x_1 \\ x_2 \\ x_3 \end{array} \right] $$
Hint:
$$ax^2+2bxy+2cxz+\cdots=a\left(x+\frac bay+\frac caz\right)^2-\frac{b^2}ay^2-\frac{c^2}az^2+\cdots$$