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Quadratic inequality

  1. Feb 15, 2015 #1
    1. The problem statement, all variables and given/known data
    Dear Mentors and PF helpers,

    Here's the question:

    The roots of a quadratic equation $$3x^2-\sqrt{24}x-2=0$$ are m and n where m > n . Without using a calculator,

    a) show that $$1/m+1/n=-\sqrt{6}$$
    b) find the value of 4/m - 2/n in the form $$\sqrt{a}-\sqrt{b}$$
    2. Relevant equations
    Sum of roots: m+ n= $$\sqrt{24}/3=2\sqrt{6}/3$$
    Product of roots = -2/3

    3. The attempt at a solution
    For a):
    I was able to show it:
    $$1/m+1/n= (n +m)/mn$$

    For b):
    My method seem to be quite long, I did simultaneous equations to solve for m and n. Using the quadratic formula. There are 2 answers for both m and n. So I choose the set of m and n that fits the criteria.
    $$m=(\sqrt{6}+\sqrt{12})/3$$
    $$n=(\sqrt{6}-\sqrt{12})/3$$

    Therefore 4/m -2/n = $$3\sqrt{12}-\sqrt{6}$$

    My answers are correct but I wonder is there a shorter way to do part (b)

    Thanks for your time
     
  2. jcsd
  3. Feb 15, 2015 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    (a) and (b) together allow to calculate 1/m and 1/n in an easy way. No matter which approach you use, it is at most one step away from finding m and n.
    You could find solutions of 1/x, that might save one or two steps, but I don't see a solution that avoids solving a quadratic equation.
     
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