Question

prove that 3+2√5 is irrational​

Answer 1

Step-by-step explanation:

We have to proof 3+2√5 is irrational​.

So, let us assume the opposite,

i.e. 3+2√5 is rational​

Hence, 3+2√5 can be written in the form $\frac{a}{b}$ where a and b (b$\neq$0) are co-prime (no common factor other than 1)

Hence, 3+2√5=$\frac{a}{b}$

2√5=$\frac{a}{b}$-3

2√5=$\frac{a-3b}{b}$

√5=$\frac{1}{2}*\frac{a-3b}{b}$

√5=$\frac{a-3b}{b}$

Here, $\frac{a-3b}{b}$ is a rational number

But √5 is irrational

Since, Rational $\neq$ Irritational

This is a contradiction

∴Our assumption is incorrect

Hence 3+2√5 is irrational​

Hence proved.