Question

Prove that 6-2√5 is irrational

Answer 1

Answer:

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let us assume that 6 -2 root5 is a rational number.

Therefore ,

6 -2root 5 = p / q where p and q are integers and q ia not equal to 0.

6 - 2root 5 = a / b where a and b are co-prime .

$\sqrt{5} = \frac{a}{2b} + 6$

Therefore ,

This contradicts the fact that root 5 is rational .

This is because of our weong assumption that 6 - 2root 5 is rational .

Therefore 6 - 2 root 5 is irrational

Answer 2

Answer:6+2√5 is irrational

Step-by-step explanation:

To the contrary, lets assume that 6+2√5 is rational ,i.e, 6+2√5 = p/q, q not equal to zero, p & q are co-primes

2√5 = p/q - 6
2√5 = p-6q/q
√5 = p-6q/2q

since, p and q are co-primes,
therefore,
p-6q/2q is rational

so,
√5 is also rational.
But, this contradicts the fact that p and q are co-primes.
This contradiction has arisen because of our incorrect assumption that 6+2√5 is rational.
So,
we conclude that 6+2√5 is irrational