How do you use the quadratic complement?

I hope you still know the binary formula (a + b)2 = a2 + 2*a*b + b2 (*).

The aim of the square supplement is to create a square polynomial term

g(x) = a*x2 + b*x + c

into a binomus and into a rest that does not contain x. For the sake of simplicity (Mathematicians call this “without restriction of the generality” or, above all, of which a has already been excluded, so we have a term

h(x) = x2 + p*x + q (**)

have. In order to add the subterm x2 + p*x to a binomin (therefore the term “addition”), it must first be made from the p a 2*(p/2):

x2 + p*x = x2 + 2*(p/2)

Now (p/2)2 is “added” and we receive

x2 + 2*(p/2)*x + (p/2)2 = (x + p/2)2 (***)

So we have found the value with the “added”. Now comes a trick that will always be important. We re-form the term in (**) by using a +0 and we do so by adding the subterm p/2 – p/2) (= 0) to (**). Consider yourself why this does not change the value of (**):

h(x) = x2 + p*x + q = x2 + p*x + p/2 – p/2 + q = x2 + 2*(p/2) + p/2 – p/2 + q

Now we can use the relationship found in (***) and finally get

h(x) = x2 + 2*(p/2) + p/2 – p/2 + q = (x + p/2)2 – p/2 + q

We have thus achieved the objective set out above. We have a binome (x+p/2)2 and a subterm that no longer contains x – p/2 + q

Why is that useful? The equation x2+p*x+q=0 can first be solved with simple deformations. These deformations lead to the so-called pq formula (and if a a is present to the so-called abc or midnight formula). Furthermore, the so-called apex of the parabola h(x) can be easily read from the equation of the square addition, namely, it lies with

xs = -p/2, ys = -p/2 + q

xs is precisely the value at which the term within the bracket becomes 0, ys is the lowest or highest point of the parabola (depending on whether a > 0 or a < 0).

First try to understand the calculation steps. But then I can only give you the recommendation to learn the procedure stubbornly and not to draw you back to a “why must I do it”. Because that only leads to confusion and frustration.