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zmudz

  • one year ago

Let \(f\) be a function such that \(\sqrt {x - \sqrt { x + f(x) } } = f(x) , \) for \(x > 1\). In that domain, \(f(x)\) has the form \(\frac{a+\sqrt{cx+d}}{b},\) where \(a,b,c,d\) are integers and \(a,b\) are relatively prime. Find \(a+b+c+d.\)

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  1. dan815
    • one year ago
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    |dw:1441598718061:dw|

  2. dan815
    • one year ago
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    |dw:1441598909212:dw|

  3. freckles
    • one year ago
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    I got a=-1 and b=2 and 2c+d=5 ... I plugged into x=2 into both forms to get this I bet you can plug in another value for x to get another equation relating c and d and then solve that system for c and d

  4. dan815
    • one year ago
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    just 1 question.. -2,1 is relatively prime im assuming is 2,-1 also said to be relatively prime

  5. freckles
    • one year ago
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    the gcd(2,-1)=1 so 2 and -1 are relatively prime

  6. dan815
    • one year ago
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    oh okay that cool, i didnt realize gcd's protected u from negative factors till now lol

  7. freckles
    • one year ago
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    plugging in x=3 will give you an easy solution to play with

  8. freckles
    • one year ago
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    well equation relating c and d I mean

  9. freckles
    • one year ago
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    gcd(x,y)=d where d>=0

  10. freckles
    • one year ago
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    if x and y are negative you can just look at the absolute values of them

  11. freckles
    • one year ago
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    \[\gcd(2,-1)=\gcd(|2|,|-1|)=\gcd(2,1)\]

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