SOLUTION: Find the rational numbers a and b given that a-2b + sqr(a+b) = sqr(6)/2 I really appreciate any help! Thanks!

Algebra ->  Rational-functions -> SOLUTION: Find the rational numbers a and b given that a-2b + sqr(a+b) = sqr(6)/2 I really appreciate any help! Thanks!      Log On


   

Question 1023321: Find the rational numbers a and b given that
a-2b + sqr(a+b) = sqr(6)/2
I really appreciate any help! Thanks!
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I%5B1%5D=sqrt%286%29%2F2 is an irrational number,
so a-2b%2Bsqrt%28a%2Bb%29 must be irrational too.
Since a and b are rational,
R=a-2b must be rational.
Since a-2b%2Bsqrt%28a%2Bb%29 is irrational, and
R=a-2b is rational,
I%5B2%5D=sqrt%28a%2Bb%29 must be irrational.
What we have in a-2b%2Bsqrt%28a%2Bb%29=sqrt%286%29%2F2 is
R%2BI%5B2%5D=I%5B1%5D .
The only way that can be possible is
system%28R=0%2CI%5B2%5D=I%5B1%5D%29 , which means
system%28a-2b=0%2Csqrt%28a%2Bb%29=sqrt%286%29%2F2%29-->system%28a=2b%2Ca%2Bb=6%2F4%29-->system%28a=2b%2C2b%2Bb=3%2F2%29-->
system%28a=2b%2C3b=3%2F2%29-->system%28a=2b%2Cb=1%2F2%29-->highlight%28system%28a=1%2Cb=1%2F2%29%29