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Is the discriminant, of the quadratic equations, the difference between the two roots

  1. Oct 11, 2011 #1
    Is the discriminant, of the quadratic equations, the difference between the two roots?
    Or is it a special case?
     
  2. jcsd
  3. Oct 11, 2011 #2

    Hootenanny

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    Re: Is the discriminant, of the quadratic equations, the difference between the two r

    I'm not sure what your asking here, but the quadratic discriminant is [itex]\Delta = b^2 - 4ac[/itex]. The two roots are

    [tex]x_\pm = \frac{-b\pm\sqrt{\Delta}}{2a},[/tex]

    with the difference being

    [tex]x_+ - x_- = \frac{-b + \sqrt{\Delta} + b +\sqrt{\Delta}}{2a} = \frac{\sqrt{\Delta}}{a}.[/tex]

    So in general, the discriminant is not the difference between the two roots. The condition for the discriminant to be the difference between the two roots is

    [tex]\Delta = \frac{\sqrt{\Delta}}{a}\text{ or } \Delta = 0\;, \Delta = a^{-2}.[/tex]

    The first corresponds to the case when you have repeated roots (obviously) and the second occurs when [itex]a^2b^2 - 4a^3c - 1 = 0[/itex].
     
  4. Oct 11, 2011 #3
    Re: Is the discriminant, of the quadratic equations, the difference between the two r

    Yes Hooteny .

    Let x and y be the two distinct roots of quadratic equation ax2+bx+c = 0
    and D = b2-4ac then xy (Product of two roots)= c/a and x+y (Sum of two roots) = -b/a .

    So we can also write a quadratic equation in this form :

    x2+bx/a+c/a = 0
    or

    A quadratic equation is written in this form :
    x2 - (Sum of two roots)x + (Product of two roots) = 0

    The only relation which establishes between equal roots of two different quadratic equations are :

    c1/a1 = c2/a2 = ... = cn/an

    and

    -b1/a1 = -b2/a2 = .... = -bn/an



    As Hooteny marks :

    Difference of two roots of a quadratic equation is : sqrt(D)/a which is not equal to D. Discriminant (D or Δ) or determinant just determines the nature of roots of a quadratic equation.
     
  5. Oct 11, 2011 #4
    Re: Is the discriminant, of the quadratic equations, the difference between the two r

    Thank you very much.
     
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