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Homework Help: What is the discriminant for this quadratic?

  1. Apr 22, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that for all real values of a and b , the roots of the eqn : ax^2-(2a+b)x+b-5a=0
    are real and different roots

    2. Relevant equations
    discriminat=b^2-4ac
    where a is the x^2 coefficient and b is the x coefficient and c is the absolute term


    3. The attempt at a solution

    (2a+b)^2 - (4a(b-5a)) = 4a^2+b^2+4ab - 4ab+20a^2 = 4a^2+b^2+20a^2

    so in order to solve the problem 4a^2+b^2+20a^2 should be > zero
    and of course 4a^2+b^2+20a^2 >=0

    but if a and b are zero then whole expression is gonna be zero thus the roots are real but the same

    so can u help me??
    Thanks
     
  2. jcsd
  3. Apr 22, 2010 #2

    ehild

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    If both a and b are zero, the original equation says that 0x^2+0x=0, an identity, which is fulfilled by any x. Mention this when you solve the problem and then say, that in any other cases the discriminant is positive (why?).

    ehild
     
    Last edited: Apr 22, 2010
  4. Apr 22, 2010 #3
    I don't really understand this.

    Because we use the square of a and b so can't be negative
    my problem is when they are zero
    Thanks
     
  5. Apr 22, 2010 #4

    ehild

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    Sorry, silly me. I wanted to say that the original equation becomes 0*x^2 + 0*x =0, an identity, of which all numbers are roots. So there is not only one root in this case or two equal ones, but infinite. You are right, this case should have been excluded from the problem.

    ehild
     
  6. Apr 23, 2010 #5
    Do u mean that there's somethin wrong with the problem??
    I think it should be : ax^2-(2a+b)x+b-5=0 instead of ax^2-(2a+b)x+b-5a=0
    Right?
     
  7. Apr 23, 2010 #6

    ehild

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    No, that equation would not have any roots for a=0 b=0. The problem should say, that "Prove that for all real, nonzero values of a and b a, the roots of the eqn : ax^2-(2a+b)x+b-5a=0 are real and different".

    ehild
     
  8. Apr 24, 2010 #7
    It would have infinite roots as you mentioned before right?

    anyways there's sometythin wrong with the problem.
    Thanks
     
  9. Apr 24, 2010 #8

    ehild

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    I meant that ax^2-(2a+b)x+b-5=0 would not have any roots for a=0, b=0, as it would look: 0*x^2-0*x-5 =0, that is -5=0 which is false, there is no x that makes it true. The original equation has any number as root.

    The formulation of the problem is wrong.

    ehild
     
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