Look at the following definition of an ordinary double point (node). The source is the book:"Freitag, Kieh - Etale Cohomology and the Weil Conjectures:"
I don't understand the geometry behind the definition. I image a node as a point with multiple tangent spaces, but I can't see this from the above conditions. In particular I'd like suggestions about the utility of completions in algebraic geometry.
Thanks in advance.
If one has an algebraic curve defined by a polynomial $F(x,y)=0$, then it is possible to solve this polynomial for $y=f(x)$ however $f(x)$ is in general a power series in $x^{\frac{1}{n}}$. This is why they are taking completions, to get infinite power series.
The function in the question $f=Q+P$ has $Q$ nonsingular quadratic so it factors into the product of two linear terms, these give the two tangents to the double point.
PS I am impressed you got so far in that book I cannot make any headway in it myself.