Let $$x^2+y^2+z^2+2xz+4y+2z+3=0$$ the equation of a quadric. Give an affine classification.

### Development:

First, we calculate the main matrix associated with the equation of the quadric and the associated characteristical polynomial.

In fact, see that $$$A=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{bmatrix} \Rightarrow det(A-xI)=-x^3+3x^2-2x=-x(x^2-3x+2)=$$$ $$$=-x(x-1)(x-2)$$$ Therefore, in view of the results, we see that there are two non-zero eigenvalues and one whose value is zero. The equation of the quadric is converted into $$$q(x,y,z)=x^2+2y^2+4y+2z+3=0$$$ Since we only have linear term for y, we complete squares for this coordinate. Finally, the equation of the quadric becomes $$$q(x,y,z)=x^2+2(y+1)^2+2z+1 \approx x^2+2y^2+2z+1=0$$$ Therefore, the limited form is of the parabolic type. Finally, we are going to find the canonical equation.

Let $$a=1$$ and $$b=\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}$$ be two real values, then the quadric is of the form $$$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+2z+1=0$$$ and this is an elliptical paraboloid.

### Solution:

The canonical equation is $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+2z+1=0$$ and this is an elliptical paraboloid.