prove that , if a,b,c and d are positive rationals such that, a+broot is equal to c+d root , then either a is equal to c and b is equal to d or b and d are squares of rationals
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no no answer for that
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Hi!
if a=c
we have a+√b=c+√d
=> √b=√d [ cancelled out a and c since both r equal]
=> b=d [squared both the sides]
now let a is not equal to c then exist a rational y such that
a=c+y
we have a+√b=c+√d
=> c+y+√b=c+√d [putting value of a]
=> y+√b=√d [cancelled out c] -----(1)
=> (y+√b)²=d [squaring both the sides ]
=>y²+b+2y√b =d
=> 2y√b = d-y²-b
=> √b= ( d-y²-b)/2y
=> √b is a rational [since d,b,y are rationals]
b is square of rational
from equation (1) we have
y+√b=√d
=> √d is a rational since y is rational and √b has been proven rational]
=> d is square of positive rational
hence, either a = c and b = d or b and d are square of rationals.
HOPE IT HELPS,!
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