Question

Prove that 3 + 2√5 is irrational.

Answer 1

⭐⭐⭐⭐⭐ ANSWER ⭐⭐⭐⭐⭐

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➡ First of all, we have to prove that √5 is an irrational number.

➡ For that, let √5 be rational number, then,

➡ √5 = p/q (where p and q are integers, q≠0 and p, q have no common factors except 1)

➡ Then, 5 = p²/q² => p²=5q²

➡ Clearly, 5 divides p.

➡ Let p = 5 m (where m is an integer)

➡ Then, p² = 5q² => 25 m² = 5 q² => 5 m² = q²

➡ Clearly 5 divides q.

➡ Thus, both p and q have a common factor 5, which contradicts that p and q have no common factors except 1.

➡ Hence, √5 is an irrational number [PROVED]


➡ Now, let 3+2√5 = r (rational number)

➡ Then, 3+2√5 = r
=> 2√5 = r-3
=> √5 = (r-3)/2 (where, √5 is irrational).

➡ Hence, our assumption was wrong and 3+2√5 is an irrational number. [PROVED]



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⭐⭐⭐ ALWAYS BE BRAINLY ⭐⭐⭐

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Answer 2

#RAM RAM ji ❤☺

3 +2√5

let 3 +2√5 be rational

3+2√5 =p/q

2√5=p/q-3/1

√5 =p-3q/2q



where √5 is rational but it's contradiction so, √5 is irrational so, our assumption is wrong .


Hence ,3 + 2√5 is irrational.


THANKS ✌☺


#NAVI ❤ HARYANVI ♠