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A question about polynomials of degree 2

  1. Jul 25, 2014 #1
    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 postive or negative number?

    Sorry if my explanation sucks but hope you understand what I mean! Thanks!
     
  2. jcsd
  3. Jul 25, 2014 #2

    AlephZero

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    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.
     
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