Adjust when breqn breaks a line in a long equation

Question

I'm using the breqn package for the first time, and while in one equation I get, using

\begin{dmath*}
\sum_{k=0}^{n} \left(\frac{k}{n} - x\right)^2 \binom{n}{k} x^k (1 - x)^{n - k}  
= \left(1 - \frac{1}{n}\right)x^2 + \frac{1}{n}x - 2x^2 + x^2 
= \frac{1}{n}x(1 - x) 
\leq \frac{1}{4n}\ ,
\end{dmath*}

something like that

I get just in the next line, something like that

\begin{dmath*}
\sum_{k \in F}\binom{n}{k}x^k(1 - x)^{n -k} 
\leq \frac{1}{\delta^2}\sum_{k \in F} \left(\frac{k}{n} - x\right)^2
\binom{n}{k}x^k (1 - x)^{n -k}
\leq \frac{1}{\delta^2}\sum_{k = 0}^{n} \left(\frac{k}{n} - x\right)^2
\binom{n}{k}x^k (1- x^k)
\leq\frac{1}{4n\delta^2} \ ,
\end{dmath*}

which is basically the same code but looks way better than the first equation. So I wonder if there's a way to adjust the dmath enviroment to make the first equality on the first equation go to the first line, and then break on the second, making my first equation have only three lines.

Answer 1

You can force breqn to use a specific layout for your equation. Unfortunately there seems to be a bug in the package because the example fails if I comment in the last line (regardless of which relational symbol I use). So, yeah, just don't use breqn.

\documentclass{article}
\usepackage{breqn}
\begin{document}
\begin{dmath*}[layout={L}]
\sum_{k\hiderel{=}0}^{n} \left(\frac{k}{n} - x\right)^2 \binom{n}{k} x^k (1 - x)^{n - k}  
= \left(1 - \frac{1}{n}\right)x^2 + \frac{1}{n}x - 2x^2 + x^2 
= \frac{1}{n}x(1 - x)
%\leq \frac{1}{4n}
\end{dmath*}
\end{document}

Answer 2

Seeing as you want to specify where to break the lines, you might as well use align here.

Unrelated to the main question, but, it looks like you are trying to add some space before the comma? If that's the case, I'd recommend using \, to add some unbreakable thinspace before the comma.

\documentclass{article}
\usepackage{mathtools}

\begin{document}

\begin{align*}
\sum_{k=0}^{n} \left(\frac{k}{n} - x\right)^2 \binom{n}{k} x^k (1 - x)^{n - k}  &= \left(1 - \frac{1}{n}\right)x^2 + \frac{1}{n}x - 2x^2 + x^2\\
&= \frac{1}{n}x(1 - x)\\
&\leq \frac{1}{4n}\,,
\end{align*}

\end{document}

Similarly, for your 2nd equation you could use this code.

\documentclass{article}
\usepackage{mathtools}

\begin{document}

\begin{align*}
\sum_{k \in F}\binom{n}{k}x^k(1 - x)^{n -k} &\leq \frac{1}{\delta^2}\sum_{k \in F} \left(\frac{k}{n} - x\right)^2 \binom{n}{k}x^k (1 - x)^{n -k} \\
&\leq \frac{1}{\delta^2}\sum_{k = 0}^{n} \left(\frac{k}{n} - x\right)^2 \binom{n}{k}x^k (1- x^k)\\
&\leq\frac{1}{4n\delta^2},
\end{align*}

\end{document}