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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
Dear Sunny Given a + √b = c + √dCase (i): Let a=c⇒ a + √b = c + √d becomesa + √b = a + √d⇒ √b = √d∴ b = dCase (ii): Let a ≠ cLet 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 = √dLet 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) RegardsArun (askIITians forum expert)
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