Question

If a, b, c, d be positive rationals such that a +√ = + √ , then prove that either a=c and b = d or b and d are squares of rationals

Answer 1

${ \green { \underline {\underline{\huge\mathbb \pink{ANSWER:-}}}}}$

${ \underline{ \underline\mathtt \blue{Given :-}}}$

Case(i): Let a=c

⇒ a + √b = c + √d becomes

a + √b = a + √d

⇒ √b = √d

∴ b = d

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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

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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

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