## 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 $$a+b+c+d.$$

1. dan815

|dw:1441598718061:dw|

2. dan815

|dw:1441598909212:dw|

3. freckles

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

just 1 question.. -2,1 is relatively prime im assuming is 2,-1 also said to be relatively prime

5. freckles

the gcd(2,-1)=1 so 2 and -1 are relatively prime

6. dan815

oh okay that cool, i didnt realize gcd's protected u from negative factors till now lol

7. freckles

plugging in x=3 will give you an easy solution to play with

8. freckles

well equation relating c and d I mean

9. freckles

gcd(x,y)=d where d>=0

10. freckles

if x and y are negative you can just look at the absolute values of them

11. freckles

$\gcd(2,-1)=\gcd(|2|,|-1|)=\gcd(2,1)$