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Quadratic problem

  1. Apr 25, 2016 #1
    1. The problem statement, all variables and given/known data

    If roots of the equation ##x^2 - 10cx - 11d = 0## are ##a, b## and those of ## x^2 - 10ax - 11 b = 0## are ##c, d## then the value of ##a + b + c + d## is (##a, b, c## and ##d## are distinct numbers)


    2. Relevant equations


    3. The attempt at a solution
    ##a+b=10c##
    ##c+d=10a##
    ## ab=-11d##
    ##cd=-11b##

    Four equations, four unknowns. Obviously, this isn't supposed to be solved using regular elimination. It gets way too terrible. I can't think of a better way though. Please help.
     
  2. jcsd
  3. Apr 26, 2016 #2

    Math_QED

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    10(a + c)
     
  4. Apr 26, 2016 #3

    ehild

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    @Math_QED is right, the sum is 10(a+c). So isolate b and d from the first two equations and substitute into the third and fourth. See what you get for a+c.
     
  5. Apr 26, 2016 #4
    Thank you!
    b = 10c - a
    d = 10a - c
    a(10c - a) = -11d
    c(10a - c) = -11b
    c2 - a2 = 11(b - d)
    a + c = 121
    b + d = 9 (a + c)
    So a + b + c + d = 1210
     
  6. Apr 26, 2016 #5

    ehild

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    You can divide the equation with c-a as a,b,c,d are all different numbers.
    Well done!
     
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