Let a, b, c, d be positive rationales such that
Then show that either a=c and b=d or b and d are square of rationals.
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Answer:
b and d are square of rationals!
Step-by-step explanation:
(i)
Let a = c:
Given: a + √b = c + √d.
⇒ a + √b = a + √d
⇒ √b = √d
On Squaring both sides, we get
⇒ b = d.
(ii)
Let a ≠ c and a = c + p{p is some rational number}
⇒ a + √b = c + √d
⇒ c + p + √b = c + √d
⇒ p + √b = √d
On Squaring both sides, we get
⇒ (p + √b)² = √d)²
⇒ p² + b + 2p√b = d
⇒ 2p√b = d - p² - b
⇒ √b = (d - p² - b/2p]
∴ Hence √b is a rational number.
This is possible only when b is a square of rational number.
Hence, d is also a square of rational number .
Hope it helps!
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