Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By Idea

Thread: Determining the smallest possible value of a in a quadratic.

  1. #1
    Junior Member
    Joined
    Sep 2016
    From
    Muskoka
    Posts
    60

    Determining the smallest possible value of a in a quadratic.

    Numbers a, b and c form an arithmetic sequence if b − a = c − b. Let a, b, c be positive
    integers forming an arithmetic sequence with a < b < c. Let f(x) = ax^2 + bx + c. Two distinct
    real numbers r and s satisfy f(r) = s and f(s) = r. If rs = 2017, determine the smallest possible
    value of a.

    I tried changing b and c into a+d and a+2d to get (r-s)((r+s+1)a+(d+1))=0 but I am stuck on how to show the least possible value of a.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    1,003
    Thanks
    509

    Re: Determining the smallest possible value of a in a quadratic.

    we need two equations

    $$f(r)-f(s)=s-r$$

    and

    $$f(r)+f(s)=s+r$$

    As you mentioned the first equation gives

    $$1+b+a r+a s=0$$

    or letting $t=r+s$,

    $$1+b+a t=0$$

    The second one can also be written in terms of $t$

    $$a (t^2 - 4034) + b t - t + 2 c=0$$

    Now eliminate $t$ in these two equations and substitute $b=a+d$ and $c=a+2d$ to get

    $$d=\frac{-1-a+2016 a^2}{1+2 a}$$

    Since $d$ is an integer, there are finitely many values possible for $a$

    {9,26,503}
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Determining Quadratic Function for a Graph
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Apr 29th 2014, 12:52 PM
  2. Replies: 19
    Last Post: Apr 20th 2013, 05:37 PM
  3. Replies: 3
    Last Post: Sep 30th 2009, 10:20 PM
  4. Replies: 2
    Last Post: Mar 26th 2009, 06:58 PM
  5. Determining Quadratic Rules
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Mar 23rd 2009, 06:12 AM

/mathhelpforum @mathhelpforum