Question

prove that 3-2√5 is irrational​

Answer 1

Step-by-step explanation:

Prove 3+2

5 is irrational.

→ let take that 3+2

5 is rational number

→ so, we can write this answer as

⇒3+2

5 = ba

Here a & b use two coprime number and b

=0

⇒25 = b

a −3⇒25 = b

a−3b

∴

5 = 2b

a−3b

Here a and b are integer so

2b

a−3b is a rational number so

5 should be rational number but

5 is a irrational number so it is contradict

- Hence 3+2

5 is irrational

Answer 2

Step-by-step explanation:

Let us assume, to the contrary, that (3-2√5) is rational.

Then, 3-2√5 = $\frac{<strong>a</strong>}{<strong>b</strong>}$. [where a&b are co-primes and b≠0]

⇒ -2√5 = $\frac{<strong>a</strong>}{<strong>b</strong>}$-3

⇒ -2√5 = $\frac{<strong>a-3b</strong>}{<strong>b</strong>}$.

⇒ √5 = $\frac{<strong>a-3b</strong>}{<strong>-2b}</strong>$.

Since, a & b are integers, so $\frac{<strong>a-3b</strong>}{<strong>-2b</strong>}$ is rational.

Thus, √5 is rational.

Bu,t this contradicts the fact that √5 is irrational. So, our assumption is incorrect.

Hence, (3-2√5) is irrational.