Question

Prove that 3 + 2√5 is irrational.​

Answer 1

Answer:

let's

2 + 3√5 be a rational no.

so , 2 + √5 = a/b

√5 = a/b - 2

( a/b - 2 ) is an rational no .

so, √5 should be a rational no.

A fact has generated √5 is irrerational no

our asuption is wrong .

so , 2+√5 is an irrerational no

Answer 2

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Let 3 + 2√5 be a rational number.

Then the co-primes x and y of the given rational number where (y ≠ 0) is such that:

$3 + 2√5 = x/y$

Rearranging, we get,

$2√5 = (x/y) – 3$

$√5 = 1/2[(x/y) – 3]$

Since x and y are integers, thus,$1/2[(x/y) – 3]$ is a rational number.

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

Thus, our assumption that $3 + 2√5$is a rational number is wrong.

Hence, $3 + 2√5$is irrational.