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Prove that 3 + 2√5 ...

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# Prove that 3 + 2√5 is irrational.

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Prove that 3 + 2√5 is irrational.

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Let us assume 3 + 25 is rational.

Then we can find co-prime x and y (y ≠ 0) such that 3 + 2√5 = x/y

Rearranging, we get,

$$2 \sqrt{5} = \frac{x}{y}-3$$

$$\sqrt{5} = \frac{1}{2}(\frac{x}{y} - 3)$$

Since, x and y are integers, thus,

$$\frac{1}{2}(\frac{x}{y} - 3)$$

is a rational number.

Therefore, 5 is also a rational number. But this contradicts the fact that 5 is irrational.

So, we conclude that 3 + 25 is irrational.

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