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Prove that (2 √3 – 1) is irrational.

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Let x = 2 √3 – 1 be a rational number. 

x = 2 √3 – 1 

⇒ x2 = (2 √3 – 1)2 

⇒ x2 = (2 √3 )2 + (1)2 – 2(2 √3)(1) 

⇒ x2 = 12 + 1 - 4 √3

⇒ x2 – 13 = - 4 √3

⇒ \(\frac{13-x^2}4\) = √3

Since x is rational number, x2 is also a rational number. 

⇒ 13 - x2 is a rational number

⇒ \(\frac{13-x^2}4\) is a rational number

⇒ √3 is a rational number 

But √3 is an irrational number, which is a contradiction. 

Hence, our assumption is wrong. 

Thus, (2 √3 – 1) is an irrational number.

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