Results 1 to 4 of 4

Math Help - further maths roots of equations

  1. #1
    Member
    Joined
    Sep 2008
    Posts
    239

    further maths roots of equations

    Hi can any body solve this question?

    The equation 2x-4x-3=0 has roots a and b. Find the quadratic equation with integer coefficients and with roots a , b
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Aug 2008
    Posts
    80
    Hello,

    Using Viete, you know the values of a+b and ab.
    Since a^3+b^3=(a+b)^3-3ab(a+b), you know the values of p=a^3+b^3 and q=a^3b^3.
    The quadratic equation with roots a^3, b^3 is x^2-px+q=0.
    Arrange it so that it has integer coefficients.

    Bye.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,686
    Thanks
    617
    Hello, djmccabie!

    The equation 2x^2-4x-3\:=\:0 has roots a and b.
    Find the quadratic equation with integer coefficients and with roots a^3,\:b^3
    The given equation is: . x^2 - 2x - \frac{3}{2} \:=\:0 with roots a and b.

    Hence: . \begin{array}{cccc}a + b &=& 2 & {\color{blue}[1]} \\ \\[-4mm] ab &=&\text{-}\frac{3}{2} & {\color{blue}[2]} \end{array}


    Cube [1]: . (a + b)^3 \:=\: 2^3 \quad\Rightarrow\quad a^3 + 3a^2b+3ab^2 + b^3 \:=\:8

    \text{We have: }\;(a^3 + b^3) + 3\underbrace{(ab)}_{-\frac{3}{2}}\underbrace{(a + b)}_{2} \;=\;8

    . . Hence: . a^3+b^3 - 9 \:=\:8\quad\Rightarrow\quad \boxed{a^3+b^3 \:=\:17}


    Cube [2]: . (ab)^3 \:=\:\left(-\frac{3}{2}\right)^3 \quad\Rightarrow\quad\boxed{ a^3b^3 \:=\:-\frac{27}{8}}


    Therefore, the equation is: . x^2 - 17x - \frac{27}{8}\:=\:0 \quad\Rightarrow\quad \boxed{8x^2 - 136x - 27 \:=\:0}

    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Sep 2008
    Posts
    239
    Thanks alot soroban! really helped i had the right idea just got a few signs wrong
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help with maths equations
    Posted in the Algebra Forum
    Replies: 2
    Last Post: November 10th 2011, 07:34 AM
  2. maths - roots of equation
    Posted in the Math Topics Forum
    Replies: 5
    Last Post: December 8th 2009, 09:15 PM
  3. further maths quadratics and roots again
    Posted in the Algebra Forum
    Replies: 2
    Last Post: September 29th 2008, 02:36 PM
  4. Maths - Parabola Equations
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 22nd 2008, 06:07 AM

Search Tags


/mathhelpforum @mathhelpforum