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Forming a quadratic equation

  1. Apr 13, 2012 #1
    The problem statement, all variables and given/known data
    Let the roots of x2 + x + 1 = 0 be [itex]\alpha[/itex] and [itex]\beta[/itex], form a quadratic equation with roots:
    (i) [itex]\alpha[/itex]2, [itex]\beta[/itex]2; and
    (ii) [itex]\frac{1}{\alpha}[/itex], [itex]\frac{1}{\beta}[/itex].


    The attempt at a solution
    sum of roots = [itex]\alpha[/itex] + [itex]\beta[/itex] = -1
    product of roots = [itex]\alpha\beta[/itex] = 1

    (i)
    sum of roots = [itex]\alpha[/itex]2 + [itex]\beta[/itex]2 = (-1)2 - 2(1) = -1
    product of roots = [itex]\alpha[/itex]2[itex]\beta[/itex]2 = 12 = 1
    Quadratic equation: x2 + x + 1 = 0

    (ii)
    sum of roots = [itex]\frac{1}{\alpha}[/itex] + [itex]\frac{1}{\beta}[/itex] = [itex]\frac{-1}{1}[/itex] = -1
    product of roots = ([itex]\frac{1}{\alpha}[/itex])([itex]\frac{1}{\beta}[/itex]) = 1
    Quadratic equation: x2 + x + 1 = 0

    Is this weird? Did I do something wrong?
    Thanks.
     
  2. jcsd
  3. Apr 13, 2012 #2

    pcm

    User Avatar

    correct.
    roots of original equations are imaginary cube roots of unity.
    which has the property that the square of first root is equal to second root,and square of second root is equal to first root.
    the same property for taking reciprocals.
     
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