16.2 Completing the square

Table 16.1: Properties of the Laplace Transform. ($k,\tau,\omega>0$, $n=1,2,\ldots$)

Completing the square simply means that you write a quadratic

$$a x^2 + b x + c$$

as

$$a \bigg[ \bigg( \underbrace{x+\frac{b}{2a}}_{\textstyl... ...rac{b^2}{4a^2}}_{\textstyle \rm {new\atop constant}} \bigg]$$

If you multiply out, you see that it is the same.

One place where you often need this is in Laplace transforms. Laplace transforms are often given in terms of a quadratic $s^2+K$ where $K$ is an arbitrary constant. But you might have $as^2+bs+c$ instead of something of the form $s^2+K$. However, you can write the part inside the square brackets above as $s^2+K$ if you use the shift theorem to account for the $b/2a$ inside the parentheses. And the additional factor $a$ is trivial to account for.