A question about polynomials of degree 2

[eric_999]

Hey!

In my calculus book they claim that a second degree polynomial always can be rewritten as x^2 - a^2 or as x^2 + a^2, if you use an appropriate change of variable. I was thinking about how this works.

Let's say we have a second degree polynomial (on the general form?) ax^2 +bx + c = 0, then I can of course rewrite it as (x + (b/2a))^2 - (b/2a)^2 + c/a = 0. My question is if they mean that (x + (b/2a)) = u, and (b/2a)^2 + c/a = k, so we always can write it like either u^2 - k^2 or u^2 + k^2 depending on if k correpsonds to a positive or negative number?

Sorry if my explanation sucks but hope you understand what I mean! Thanks!

 

[AlephZero]

That's correct. $u = x + \frac{b}{2a}$ makes the coefficient of $u$ zero. You can then write the constant term as $+k^2$ or $-k^2$ depending on whether it is positive or negative.

You might like to think about how factorizing $u^2 - k^2$ is similar to solving the quadratic equation $ax^2 + bx + c = 0$ by completing the square, and the fact that $u^2 + k^2$ has no real roots.