If x=3+2√2then find whether x+1/x is rational or irrational.
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x=3+2√2
so
1/x=1/3+2√2
1/x=1 / 3+2√2 * (3-2√2) / (3-2√2)
1/x=3-2√2 / (3)²-(2√2)²
1/x=3-2√2 / 9-8
1/x=3-2√2 / 1
1/x=3-2√2
THUS
x+1/x=(3+2√2)+(3-2√2 )=6
SINCE
x+1/x can be written in the form p/q ,where p=6 and q≠0
hence x+1/x is a rational number
so
1/x=1/3+2√2
1/x=1 / 3+2√2 * (3-2√2) / (3-2√2)
1/x=3-2√2 / (3)²-(2√2)²
1/x=3-2√2 / 9-8
1/x=3-2√2 / 1
1/x=3-2√2
THUS
x+1/x=(3+2√2)+(3-2√2 )=6
SINCE
x+1/x can be written in the form p/q ,where p=6 and q≠0
hence x+1/x is a rational number
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