Results 1 to 6 of 6
Like Tree3Thanks
  • 1 Post By HallsofIvy
  • 1 Post By MINOANMAN
  • 1 Post By Paze

Math Help - Quadratic proof

  1. #1
    Newbie
    Joined
    Feb 2013
    From
    US
    Posts
    14
    Thanks
    3

    Quadratic proof

    I was doing some problems last night, and I came across this one and I am not sure how to get started.

    If the polynomial Eq. A: ax^2+bx+c does not have any (real*) roots, prove that Eq. B: ax^2+bx-c has two (real*) roots.

    *In full disclosure, the text did not include the word 'real' before the word 'roots'. Obviously, by the Fundamental Theorem of Algebra, both will have two roots.

    I was thinking about arguing that equation A must have a discriminant that is negative (  b^2-4ac < 0 ) and therefore Equation B will have a positive discriminant (  b^2-4a(-c) = b^2+4ac > 0 ). A quadratic with a positive discriminant has two real roots.

    Is there a better / more clever way to solve this?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,693
    Thanks
    1466

    Re: Quadratic proof

    Okay, but you are skipping over the crucial part of the proof: how does b^2- 4a(-c)> 0 follow from b^2- 4ac< 0?
    Thanks from heaviside
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Feb 2013
    From
    Saudi Arabia
    Posts
    441
    Thanks
    86

    Re: Quadratic proof

    since the quadratic ax^2+bx+c=0 has no real roots b^2<4ac .and since b^2 is positive either both a and c are positive or both are negative.In the second quadratic the c is negative therefore the product 4ac becomes now negative therefore b^2<4a(-c) is not possible since b^2 is a positive number .Therefore b^2-4a(-c)>0 and in this case the quadratic has two real roots.
    Thanks from heaviside
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Feb 2013
    From
    US
    Posts
    14
    Thanks
    3

    Re: Quadratic proof

    Thank you HallsofIvy and Minoanman!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member Paze's Avatar
    Joined
    Nov 2012
    From
    Iceland
    Posts
    379
    Thanks
    19

    Re: Quadratic proof

    Hi Heaviside.

    If it is to any help to you, I have scribbled the actual proof: http://s12.postimage.org/bpozskgal/q...rmulaproof.png

    I changed letters from upper-case to lower-case. These are the same values, I just got lazy.
    Thanks from heaviside
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Feb 2013
    From
    US
    Posts
    14
    Thanks
    3

    Re: Quadratic proof

    Thanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. quadratic residue proof
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: November 26th 2012, 07:39 PM
  2. proof with quadratic residue
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: July 17th 2011, 03:17 PM
  3. [SOLVED] Quadratic Residue proof
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: July 13th 2011, 04:10 PM
  4. [SOLVED] Quadratic Proof
    Posted in the Algebra Forum
    Replies: 2
    Last Post: June 23rd 2011, 07:57 AM
  5. Quadratic Integers Proof
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: January 24th 2011, 04:49 PM

Search Tags


/mathhelpforum @mathhelpforum