Given this quadric:
$x_1^2+5x_2^2+9x_3^2+4x_1x_2+2x_1x_3+10x_2x_3-2x_3=2$
Maple screenshots:
How to put it into the normal form
$\Large\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}-\frac{x_3^2}{c^2}=1$
I've tried the standard way, namely putting it in $x^TAx+c^Tx+b=0$, then calculating (using maple) the eigenvalues of $A$, but they turned out to be very complicated terms and since it should be done without software, I'm guessing this is the wrong way to do it.
Is there any other way to transform this quadric into normal form?
Doing some completing the square will achieve it:
$$x_1^2+4x_1x_2+4x_2^2+x_2^2+9x_3^2+2x_1x_3+10x_2x_3-2x_3=2\\ \implies (x_1+2x_2)^2+x_2^2+9x_3^2+2x_1x_3+10x_2x_3-2x_3=2$$
Let $y=x_1+2x_2$, then $x_1=y-2x_2$.
$$y^2+x_2^2+9x_3^2+2(y-2x_2)x_3+10x_2x_3-2x_3=2\\ \implies y^2+2yx_3+x_3^2+x_2^2+6x_2x_3+8x_3^2-2x_3=2\\ \implies (y+x_3)^2+(x_2+3x_3)^2-(x_3+1)^2=1$$
Now change it back to the original variables:
$$(x_1+2x_2+x_3)^2+(x_2+3x_3)^2-(x_3+1)^2=1$$