Question

2. Prove that 4 - 2 √5 is an irrational number.​

Answer 1

Answer:

Let us assume, to the contrary that 4 - 5√2 is rational.

So, we can find co-prime integers a and b(b ≠ 0)

such that

$\sf4 - 5√2 = \frac{a}{b}$

$\sf \dashrightarrow \: 5√2 = 4 - \frac{a}{b}$

$\sf \dashrightarrow √2 =\frac{(4b - a)}{5b}$

Since a and b are integers, (4b - a)/5b is rational.

So, √2 is rational.

But this contradicts the fact that √2 is irrational.

Hence, 4 - 5√2 is irrational.

Answer 2

Answer:

let

4-2root5 = a/b

squaring on both the sides

(4-a/b)^2=(2rot5)^2

16+a^2/b^2-8a/b=20

a^2/b^2=20-16+8a/b

a/b=root(20-16+8a/b)