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

=

b

a

Here a & b use two coprime number and b



=0.

⇒2

5

=

b

a

−3

⇒2

5

=

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

Question:-

↗Prove that $\sf 3 - 2\sqrt{5}$ is irrational.

Answer:-

↗ Let us assume to the contrary that $3 -2 \sqrt{5}$ is a rational number

↗Then, it can be expressed in the form $\sf \dfrac{a}{b}$, where a,b are integers and b $\neq$ 0.

↗Now, $\sf 3 - 2\sqrt{5} = \dfrac{a}{b}$ , where a,b are co-primes and b $\neq$ 0.

↗On reaarranging , we get.

➡$\sf 2\sqrt{5} = \dfrac{a}{b} - 3$

↗Since, a and b are integers and $\sf \dfrac{a}{2b} , \dfrac{3}{2}$ are rationak number .

↗Therfore $\sf \dfrac{a}{2b} - \dfrac{3}{2}$ is a rational number. ( Since diffrence of two rational number is also a rational number.)

↗$\sqrt{5}$ is a rational number.

↗But $\sqrt{5}$ is an irrational number.

↗This shows that our assumption is incorrect

↗So $\sf 3 - 2 \sqrt{5}$ is irrational.

Hence, proved.