Question

Find the number that we can place in the box, so that the resulting quadratic is the square of a binomial.

Answer 1

We need a number to complete

$(x+c)^2=x^2+2xc+c^2$

Since we have

$2xc=\dfrac{4}{5}x \iff 2c=\dfrac{4}{5}\iff c=\dfrac{4}{10}=\dfrac{2}{5}$

And so the expanded square is

$\left( x+\dfrac{2}{5}\right)^2=x^2+\dfrac{4}{5}x+\dfrac{4}{25}$