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Given that √3 is irrational, prove that 5 + 2√3 is irrational.

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Solution:

Let us assume 5 + 2√3 is rational, then it must be in the form of p/q where p and q are co-prime integers and q ≠ 0 

i.e 5 + 2√3 = p/q 

So √3 = p−5q/2q ……………………(i) 

Since p, q, 5 and 2 are integers and q ≠ 0, HS of equation (i) is rational. But LHS of (i) is √3 which is irrational. This is not possible. 

This contradiction has arisen due to our wrong assumption that 5 + 2√3 is rational. So, 5 + 2√3 is irrational.

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