Question

prove that 3 + 2 root 5 is irrational given that root 5 is irrational​

Answer 1

Step-by-step explanation:

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=>3+2√5

Let, 3+2√5 be rational

=> 3+2√5 = p/q (where p and q are co-prime & q is not equal to 0)

3+2√5 = p/q

squaring both sides,

(3+2√5)^2 = (p/q)^2

9+4×5 = p^2/q^2

29 = p^2/q^2

29×q^2 = p^2 ..............................................................eq.1

=> the above eq.1 shows that p^2 is factor of 29

=> p is also factor of 29

p= 29m (where m is some integer)................................eq.2

3+2√5 = p/q

from eq.2 we get,

3+2√5 = 29m/q

squaring both sides,

9+4×5 = 841m^2/ q^2

29 = 841m^2/q^2

q^2 = 841m^2/29

q^2 = 29m^2...............................................................eq.3

=> eq.3 shows that q^2 is also a factor of 29

but co-prime number have no factors

=> 3+2√5 is an irrational number.

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