3x2−2xy+3y2+4x+4y−4=0 canonical form

Given equation of the surface of 2-order:
$$- 2 x y + 4 x + 3 x_{2} + 4 y + 3 y_{2} - 4 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x y_{2} + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y y_{2} + 2 a_{24} y + a_{33} y_{2}^{2} + 2 a_{34} y_{2} + a_{44} = 0$$
where
$$a_{11} = 0$$
$$a_{12} = -1$$
$$a_{13} = 0$$
$$a_{14} = 2$$
$$a_{22} = 0$$
$$a_{23} = 0$$
$$a_{24} = 2$$
$$a_{33} = 0$$
$$a_{34} = \frac{3}{2}$$
$$a_{44} = 3 x_{2} - 4$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22} + a_{33}$$

     |a11  a12|   |a22  a23|   |a11  a13|
I2 = |        | + |        | + |        |
     |a12  a22|   |a23  a33|   |a13  a33|

$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$

     |a11  a14|   |a22  a24|   |a33  a34|
K2 = |        | + |        | + |        |
     |a14  a44|   |a24  a44|   |a34  a44|

     |a11  a12  a14|   |a22  a23  a24|   |a11  a13  a14|
     |             |   |             |   |             |
K3 = |a12  a22  a24| + |a23  a33  a34| + |a13  a33  a34|
     |             |   |             |   |             |
     |a14  a24  a44|   |a24  a34  a44|   |a14  a34  a44|

substitute coefficients
$$I_{1} = 0$$

     |0   -1|   |0  0|   |0  0|
I2 = |      | + |    | + |    |
     |-1  0 |   |0  0|   |0  0|

$$I_{3} = \left|\begin{matrix}0 & -1 & 0\\-1 & 0 & 0\\0 & 0 & 0\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}0 & -1 & 0 & 2\\-1 & 0 & 0 & 2\\0 & 0 & 0 & \frac{3}{2}\\2 & 2 & \frac{3}{2} & 3 x_{2} - 4\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda & -1 & 0\\-1 & - \lambda & 0\\0 & 0 & - \lambda\end{matrix}\right|$$

     |0      2    |   |0      2    |   | 0      3/2   |
K2 = |            | + |            | + |              |
     |2  -4 + 3*x2|   |2  -4 + 3*x2|   |3/2  -4 + 3*x2|

     |0   -1      2    |   |0   0       2    |   |0   0       2    |
     |                 |   |                 |   |                 |
K3 = |-1  0       2    | + |0   0      3/2   | + |0   0      3/2   |
     |                 |   |                 |   |                 |
     |2   2   -4 + 3*x2|   |2  3/2  -4 + 3*x2|   |2  3/2  -4 + 3*x2|

$$I_{1} = 0$$
$$I_{2} = -1$$
$$I_{3} = 0$$
$$I_{4} = \frac{9}{4}$$
$$I{\left(\lambda \right)} = - \lambda^{3} + \lambda$$
$$K_{2} = - \frac{41}{4}$$
$$K_{3} = - 3 x_{2} - 4$$
Because
$$I_{3} = 0 \wedge I_{2} \neq 0 \wedge I_{4} \neq 0$$
then by type of surface:
you need to
Make the characteristic equation for the surface:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
or
$$\lambda^{3} - \lambda = 0$$
$$\lambda_{1} = -1$$
$$\lambda_{2} = 1$$
$$\lambda_{3} = 0$$
then the canonical form of the equation will be
$$\tilde y2 \cdot 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
and
$$- \tilde y2 \cdot 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
$$- \tilde x^{2} + \tilde y^{2} + 3 \tilde y2 = 0$$
and
$$- \tilde x^{2} + \tilde y^{2} - 3 \tilde y2 = 0$$
$$- 2 \tilde y2 + \left(\frac{\tilde x^{2}}{\frac{3}{2}} - \frac{\tilde y^{2}}{\frac{3}{2}}\right) = 0$$
and
$$2 \tilde y2 + \left(\frac{\tilde x^{2}}{\frac{3}{2}} - \frac{\tilde y^{2}}{\frac{3}{2}}\right) = 0$$
this equation is fora type hyperbolic paraboloid
- reduced to canonical form