Let a,b,c,d be positive rationals such that a+√b=c+√d then prove that a=c and b=d or b and d are square of rationals

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Arun · Answer

Dear Sunny

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

sqrt(b) = [d - k² - b]/2k

Notice that RHS is a rational number.

Hence sqrt(b) is a rational number.

This is possible when b is a square of rational number.

Thus d is also a square of rational number as

k + sqrt(b) = sqrt (d)

Regards

Arun (askIITians forum expert)

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Let a,b,c,d be positive rationals such that a+√b=c+√d then prove that a=c and b=d or b and d are square of rationals

Let a,b,c,d be positive rationals such that a+√b=c+√d then prove that a=c and b=d or b and d are square of rationals

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