Question

Prove that 2√5 + 3 is an irrational number​

Answer 1

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Answer 2

first, we need to prove that√5 is a irrational number,

so,

Let us assume that √5 is a rational number.

Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒√5=p/q

On squaring both the sides we get,

⇒5=p²/q²

⇒5q²=p² —————–(i)

p²/5= q²

So 5 divides p

p is a multiple of 5

⇒p=5m

⇒p²=25m² ————-(ii)

From equations (i) and (ii), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5

⇒q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number.

Now,

let us assume that 2√5+3 is a rational number.

a and b are co prime numbers

2√5+3 =a/b

2 √5=a-3b/b

√5=a-3b/2b

here,

a-3b/2b is a rational number

we know that

√5 is a irrational number.

therefore,

2√5+3. is a irrational number