This is related to the question posted in:
The discriminant function in linear discriminant analysis
In the one dimensional case, where $p_k(x) = \dfrac{f_k(x)\pi_k}{P(X = x)}$, where $f_k(x) = \dfrac{1}{\sqrt{2\pi}\sigma}\exp(- \frac{1}{2\sigma^2}(x-\mu_k)^2)$.
After taking the logarithm, the discriminant function is given as:
$\delta(x) = \log \pi_k + \dfrac{x \mu_k}{\sigma^2} - \dfrac{\mu_k^2}{2\sigma^2}$.
Why is there no term involving $-\dfrac{x^2}{2\sigma^2}$? This is the first term in the expansion of $(x-\mu_k)^2$. Why is this neglected?
And now that I have a discriminant function, what is the next step?