Results 1 to 7 of 7

Math Help - common roots of two cubic equations

  1. #1
    Junior Member
    Joined
    Jun 2010
    Posts
    39

    common roots of two cubic equations

    (x^3)+(k*(x^2))+x+3=0;
    (x^3)+(2*(x^2)+kx+2=0;

    Find value of k, if the above two equations have a common root.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2012
    From
    Sheffield England
    Posts
    440
    Thanks
    76

    Re: common roots of two cubic equations

    In both coefficient of x^3 is 1. 1st cubic ends in +3 and 2nd in +2. So any common factors can only be (x+1) or (x-1) giving roots -1 or 1
    If x=1 is a common root 1+k+1+3=0 and 1+2+k+2=0 Both give k=-5
    If x=-1 is a common root -1+k-1+3=0 and -1+2-k+2=0 These dont give the same k
    So answer is k=-5 giving a common root of x=1
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,527
    Thanks
    773

    Re: common roots of two cubic equations

    Quote Originally Posted by biffboy View Post
    In both coefficient of x^3 is 1. 1st cubic ends in +3 and 2nd in +2. So any common factors can only be (x+1) or (x-1) giving roots -1 or 1
    Did you use the rational root theorem to conclude this? I believe there is another k that gives a common root, but it is irrational and hard to express...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Mar 2012
    From
    Sheffield England
    Posts
    440
    Thanks
    76

    Re: common roots of two cubic equations

    Yes I did use that theorem.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Jun 2010
    Posts
    39

    Re: common roots of two cubic equations

    Quote Originally Posted by biffboy View Post
    In both coefficient of x^3 is 1. 1st cubic ends in +3 and 2nd in +2. So any common factors can only be (x+1) or (x-1) giving roots -1 or 1
    Can you please explain this further?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Jun 2010
    Posts
    39

    Re: common roots of two cubic equations

    Quote Originally Posted by emakarov View Post
    Did you use the rational root theorem to conclude this? I believe there is another k that gives a common root, but it is irrational and hard to express...
    What is the rational root theorem? Can you please link to an explanation with examples.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,527
    Thanks
    773

    Re: common roots of two cubic equations

    Quote Originally Posted by swordfish774 View Post
    What is the rational root theorem? Can you please link to an explanation with examples.
    See Wikipedia; it has an example. In this case, if p / q is a root of the first equation, then p | 3 and q | 1 (the notation "|" mean "divides"). Similarly, if p / q is a root of the second equation, then p | 2 and q | 1. Therefore, p and q are 1 and the only possible rational roots are 1 and -1.

    An alternative way is to subtract the second equation from the first one to get a quadratic equation and solve it for x. It is not as scary as it seems. This gives the solution shown above as well as another root as a function of k. Substituting this root into one of the equations gives a cubic equation on k, but its root is apparently irrational.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. roots of cubic equations
    Posted in the Algebra Forum
    Replies: 4
    Last Post: May 6th 2012, 10:04 PM
  2. [SOLVED] Quadratic and cubic equation -show that -(common roots)
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: August 19th 2011, 11:47 PM
  3. Roots of cubic equations and new equation
    Posted in the Algebra Forum
    Replies: 1
    Last Post: January 13th 2010, 02:46 AM
  4. Equations with Common Roots
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: August 25th 2009, 03:12 AM
  5. Finding the roots of cubic equations
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: April 18th 2009, 07:16 PM

Search Tags


/mathhelpforum @mathhelpforum