Question

HELP PLZ! IAM CRYYING LOOK AT MY STUFF, STOP MY TEARS!
Okay basically I have an equation with 3 variables and highest degree is 2, I want to rewrite it in a new basis where its form has only the 2 degree terms!

GOAL:
To change

ax^2+by^2+cz^2+dxy+exz+fyz+gx+hy+iz = d
into the form
X^2+Y^2+Z^2=D


Converting Quadratic Forms
f(X_1,X_2,X_3)=\[ [c1,c2,c3,...,c10]\\ dot [X_1^2+X_2^2+X_3^2+X_1X_2+X_2X_3+X_1X_3+X_1+X_2+X_3+1]\]
The picture below has this written out in matrix form

convert into form
f(X_1,X_2,X_3)=g(Y_1,Y_2,Y_3)=\[k_1Y_1^2+k_2Y_2^2+k_3Y_3^2\]

where Y_1,Y_2,Y_2 = f(X_1,X_2,X_3) some linear basis transformation