# How is Big-O notation used in equalities?

I recently learned about big-O notation and I think I get it but in some uses it does not line up with what I think I understand it as. In the wikipedia page it calls using the form $f(x) = O(g(x))$ an abuse of notation because all it is saying is that there is a relationship between the two. It goes on to use similar notation to say

$g(x) = h(x) + O(f(x))$

expresses the same as

$g(x) - h(x) = O(f(x))$

This is easy enough to understand, but I fail to take valuable meaning from statements such as

$(n+O(n^{1/2}))(n + O(\log n))^2 = n^3 + O(n^{5/2})$

where it would be impossible to set it up as an equality with my big-O statement on one side then "take away" the misleading notation. If someone could help me understand this I would greatly appreciate it.

In equalities, such as $$(n+O(n^{1/2}))(n+O(\log n))^2=n^3+O(n^{5/2})$$
The "$O(f(x))$" means "some function that is $O(f(x))$" i.e.
$$(n+f_1(n))(n+f_2(n))^2=n^3+f_3(n)$$ Where $$f_1(n)\in O(n^{1/2}),\quad f_2(n)\in O(\log n),\quad f_3(n)\in O(n^{5/2})$$
Think of an expression like $g(x) = h(x) + O(f(x))$ as shorthand for “$g(x) = h(x) + \phi(x)$, where $\phi(x)\in O(f(x))$.”
Basically, equations with big-Oh notation on both sides need to be read left to right. For example, $$(n+O(n^{1/2}))(n + O(\log n))^2 = n^3 + O(n^{5/2})$$ means that for any functions $f$ and $g$ so that $f(n)=n+O(n^{1/2})$ and $g(n)=(n+O(\log n))^2$, it is true that $f(n)g(n)=n^3+O(n^{5/2})$. (Never mind that this statement, though true, is almost ridiculously weak. A better right hand side would be $n^2+O(n^{3/2})$.)