Finding x² + y² when 1/√(4 − 2√3) = x + y√3

[wisredz]

can't get this!

Hi there,
I have a question that I cannot solve. Here it is.

$$\frac{1}{\sqrt(4-2\sqrt3)}=x+y\sqrt3$$

then what is x^2+y^2?

All I did was finding what left hand side stood for. It equals

$$\frac{\sqrt3 + 1}{2}$$

Any help?

 

[Wiz]

if i understand u right then tht implies x=y=1/2...so find wht u want...

 

[wisredz]

How does he implies that? I got there before but I supposed that x and y are not irrational

 

[marlon]

do you know the numerical solution for this problem ?

marlon

 

[marlon]

hello ? are you dead ?

 

[HallsofIvy]

If x and y are rational, then x= y= 1/2 so x²+ y²= 1/2 is the only solution. If x and y are allowed to be rational, then there are an infinite number of solutions.

 

[marlon]

how is the left hand side equal to $$\frac{\sqrt3 + 1}{2}$$ ?


marlon

 

[wisredz]

that's because of this.

suppose that a=x+y and b=xy then

$$\sqrt (a + 2\sqrt b) = \sqrt x+ \sqrt y$$

Ivy, I don't get what you mean. How do you know if the numbers x and y are rational then the only solution is x=y=0.5? and how do you know there is an infinite number of solutions if they are irrational?

 

[Wiz]

if x and y are rational then irrational terms on both sides of the eq must be equal adn also rational terms on both sides must be equal.hence y=x=0.5...get it?

 

[wisredz]

yeah I know it, I said I did it that way. But the problem is that nothing is told about it. Anyway thnaks, I think the question wasn't complete in this case