Question

Prove that 5 - √5 is irrational​

Answer 1

Answer:

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Answer^_^

let 5- √5 be rational.

so.

5-√5=a/b

√5=-( a/b-5)

√5 = -a+5b/5

since

a and b are integers so

-a+5b/b is rational

then √5 is also rational

but this contradict the fact that√5 is irrational

hence,

5-√5 is irrational.......

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Answer 2

suppose that 5- root 5 is rational.

Then, 5- root 5 = p/q ( where p and q are co primes and q not equal to 0)

$\sqrt{5 } = \frac{p}{q} + 5$

$\sqrt{5} = \frac{p + 5q}{q}$

But we know that p, 5q and q are integers.

therefore p + 5q/q is rational

But root 5 is irrational and rational is not equal to irrational.

this contradicts the fact that root 5 is rational

There fore 5- root 5 is irrational.

HOPE IT HELPS