i am trying to work out the quadratic approximation to the binomial distribution.

so far i know that log-likelihood for the probability p is the mean of x.

i believe the observed fishers information to be:
ihat = n/ (phat*(1-phat))

is the quadratic approximation:
-ihat*((p-phat)^2)*(1/2) ????

plotting this in R doesnt give the result i am expecting. the approximation doesnt join the log likelihood at the maximum.

where am i goin wrong?

thanks