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# If the sum and product of sursa a+√b and c+√d both are rational, then one of the conditions to be satisfied is  1. a=b .  2. √b + √d =0  3.  a = d .  4.  b = c

Arun
25763 Points
2 years ago
Given a + √b = c + √d
Case (i): Let a=c
⇒ a + √b = c + √d becomes
a + √b = a + √d
⇒ √b = √d
∴ b = d
Case (ii): Let a ≠ c
Let us take a = c + k where k is a rational number not equal to zero.
⇒ a + √b = c + √d becomes
(c + k) + √b = c + √d
⇒ k + √b = √d
Let us now square on both the sides,
⇒ (k + √b)2 = (√d)2
⇒ k2 + b + 2k√b = d
⇒ 2k√b = d – k2 – b

Notice that the RHS  is a rational number.
Hence √b is a rational number
This is possible only when b is square of a rational number.
Thus d is also square of a rational number as k + √b = √d.... Hope it helps