Question

Prove that,if a, b, c and d are positive rationals such that a +√b =c+√d, then either a=c and b=d or b and d are squares of rationals. ​

Answer 1

Proof : When a = c,

Then, a + √b = c + √d

→ a + √b = a + √d

→ √b = √d

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So, let a ≠ c and a = x + c.

→ a + √b = c + √d

→ x + c + √b = c + √d

→ x + √b = √d

→ (x + √b)² = d

→ d = x² + b + 2x√b

→ (d - x² - b)/2x = √b

→ b = [(d - x² - b)/2x]²

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Hence b is square of rational.

Similarly, d is also sqaure of ration.

Q.E.D