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Can all quadratic equations be factorised?

  1. Oct 25, 2011 #1
    1. The problem statement, all variables and given/known data

    Is it true that all quadratic equations (ax2+bx+c) can be factorised into this form: (ax+b)(cx+d)?

    I think so, but I'm not certain.
     
  2. jcsd
  3. Oct 25, 2011 #2

    gb7nash

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    If d is just some arbitrary number and you didn't accidentally duplicate your letters, then no.

    Example:

    y = x2-2x+1

    So a = 1, b = -2, c = 1

    So y = (x-1)(x-1)

    a, c and d are fine, but you have a -1 instead of b.
     
  4. Oct 25, 2011 #3

    SammyS

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    As gb7nash said the coefficients a, b, and c, in (ax2+bx+c) are not, in general, the same as those in (ax+b)(cx+d).

    So, can (ax2+bx+c) always be factored into the form, (px+q)(rx+s)?
    The answer is ... maybe.

    If you allow q and s to be complex numbers, then the answer is yes.

    If you restrict q and s to be real numbers, then you must have [itex]\sqrt{b^2-4ac}\ge0[/itex]

    If q and s must be rational numbers, then you must have [itex]\sqrt{b^2-4ac}[/itex] be a perfect square.​

    This all assumes that a, b, and c are rational numbers.
     
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