SOLUTION: If (x+ysqrt3)^2 = 48+ 24sqrt3, find the values of x and y, given that they are rational.

Question 1111112: If (x+ysqrt3)^2 = 48+ 24sqrt3, find the values of x and y, given that they are rational.
Answer by ikleyn(52792) (Show Source): You can put this solution on YOUR website!
.
It is straightforward.

x^2 + 3y^2 = 48
2xy        = 24    ====>  xy =  = 12.


x =    x^2 = 

 + +  = 48

144 + 3y^4 = 48y^2

3y^4 - 48y^2 + 144 = 0

y^4 - 16y^2 + 48 = 0

 =  = .


1)  y^2 =  = 12  ====>  y is irrational  ===>  the solution does not work;


2)  y^2 =  = 4  ====>  y = +/-2  ====>  x =  = +/- 6.


Answer.  Two solutions are  (x,y) = (6,2);  (x,y) = (-6,-2).

Solved.

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