First, what is the purpose of the "complete the square" model?

When examining a quadratic function such as

      (Expression 1)

in some cases would be nice to be able to factor this expression. For example, this is the case when calculating the roots of the function.

However, not all quadratic expressions can be factored.

But what is always possible, is that every quadratic polynomial can be expressed as a squared expression of x with a trailing constant, such as

              (Expression 2).

This is not a full factoring, as there is a leftover constant: k, but sometimes it's enough.

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The question becomes: How can we transition Expression 1 into Expression 2? This process is called "complete the square".

The idea behind it is that Expression 1 can be transformed easily into such that the coefficient a = 1, as follows.

This means there are equivalent coefficients we can use instead, to deal with a simpler polynomial, or the function (depending on the situation):

is the same as

Then we only have to complete the square for

     (Expression 3)

as long as we remember to keep the outer bracket that multiply a  at the front.

Let's try to fit Expression 3 into the algebraic formula of a proper square with leftover constants (like the one in Expression 2).

The proper square we rely on, which by the way Expression 3 cannot fit into without some extra constants, looks like this.

(Expression 4)

After adding a constant k to both side of the above equation we have this

(Expression 5)

And since we cannot overlap, or equate the expression we are working on with Expression 4, but we can definitely do so with Expression 5.

So let's overlap the two expressions and handle each arrow separately.

The red arrow shows an element that is already identical in both. No need to do anything.