Converting a 2nd degree polynomial to a perfect square

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Ok, I honestly couldn't come up with a better name for this lesson.

Let's do it:

We have a second degree polynomial:

and we want to find a change of variable that transforms it to a perfect square:

Of course, for the transformation to exist, the polynomial's coefficients must have 1 degree of freedom! We will assume it is so. Expanding the expression we get:

Identifying the coefficients we get:

And the condition (for which we required the degree of freedom):

Further interpretation of the condition obtained[edit | edit source]

Notice that the condition is equivalent to saying the determinant of the polynomial is equal to 0. When a polynomial's determinant was 0, the polynomial has a double root. Note this is just what we asked for when equating it to . (A double root is present at )