Question

Prove that 2-3√5 is an irrational number.​

Answer 1

Step-by-step explanation:

Let 2-3√5 is a rational number

A rational number can be written in the form of p/q where p,q are integers.

2-3√5 = p/q

Squaring on both sides,

(2-3√5)²=(p/q)²

[2²+(3√5)²-2(2)(3√5)]=p²/q²

[4+9(5)-12√10]=p²/q²

[4+45-12√10]=p²/q²

12√10=49 - p²/q²

12√10=(49q²-p²)/q²

√10=(49q²-p²)12q²

p,q are integers then (47q²-p²)/12q² is a rational number.

Then,√10 is also a rational number.

But this contradicts the fact that √10 is a rational number.

So,our supposition is false.

Hence,2-3√5 is irrational number.

Hope it helps