What does "turn into a perfect square" mean here?

Question

I'm reading a book on calculus of variations, and it says the following:

What does he mean by turning the integrand into a perfect square?

By the way, yes I have looked on wikipedia and other sites, and did not find an answer I understood.

Answer 1

A perfect square (also just called a square if it's just a number) is an expression of the form $(\text{something})^2$. Their most useful attribute is that real perfect squares are nonnegative. We can also live with $P \cdot (\text{something})^2 = (\sqrt{P}(\text{something}))^2 $, although the left-hand side is not truly a perfect square

To turn the integrand into a perfect square, complete the square on the $\eta'$ term: $$ P\eta^2 + 2w\eta\eta' + (Q+w')\eta^2 = P\left(\eta+\frac{w\eta'}{P}\right)^2 + \left( Q+w' - \frac{w^2\eta'}{P} \right)\eta^2. $$ For this to be a perfect square, the second term has to vanish, which gives you the condition.