Prove that 3 + 2√5 is Irrational

 

❓ Question

Prove that:

$$3+2\sqrt{5}$$

is irrational.

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🖼️ Solution Image

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✍️ Short Explanation

This proof uses the contradiction method.

We assume that:

$$3+2\sqrt{5}$$

is rational and then show that this assumption leads to a contradiction 💯

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🔹 Step 1 — Assume $3+2\sqrt{5}$ is Rational

Let:

$$3+2\sqrt{5}=\frac{a}{b}$$

where:

• $a$ and $b$ are integers
• $b \ne 0$
  

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🔹 Step 2 — Rearrange the Equation

$$2\sqrt{5}=\frac{a}{b}-3$$

Taking LCM:

$$2\sqrt{5}=\frac{a-3b}{b}$$

Dividing both sides by $2$:

$$\sqrt{5}=\frac{a-3b}{2b}$$

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🔹 Step 3 — Analyse the Result

Since:

• $a$ and $b$ are integers
• $a-3b$ is also an integer

Therefore,

$$\frac{a-3b}{2b}$$

is rational.

This implies:

$$\sqrt{5}$$

is rational.

But this is false because:

$$\sqrt{5}$$

is irrational.

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🔹 Step 4 — Contradiction

Our assumption is wrong.

Hence,

$$3+2\sqrt{5}$$

is irrational.

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✅ Final Answer

$$\boxed{3+2\sqrt{5}\text{ is irrational}}$$

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⭐ Key Insight

• Rational + irrational = irrational
• Contradiction method helps prove irrationality

🧠 Memory Line:

Irrational number ke saath rational add/subtract karne par result irrational hi rehta hai

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📚 Related Topics

• Proof that √5 is Irrational
• Irrational Numbers Proof Using Contradiction Method
• Rational and Irrational Numbers