Question

prove that root 5 minus 2 is an irrational number​

Answer 1

Answer:

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Step-by-step explanation:

=>> 2-√5 be a rational number

2-√5 = p/q (where p&q is a coprime intgers and q is not equal to 0)

-√5=p/2q

[√5=-p/2q ]------------(1)

-p/2q = rational number. (bcoz p & q are co prime )

-(2)

from (1)&(2)

√5 = rational number

which is not possible...

therefore our supposition of assuming 2-√5 was wrong ...

therefore it's irrational number

HENCE PROVED

Answer 2

√5-2

let √5-2 be a rational number

√5-2=x

squaring on both the sides, we get

(√5-2)^2=x^2

5+4-2×2×√5=x^2

9-4√5=x^2

9-x^2=4√5

√5=9-x^2÷4

here, x is a rational number

= x^2 is a rational number

=9-x^2 is a rational number

=9-x^2÷4 is also a rational number

√2=9-x^2÷4 is a rational number

but √2 is an irrational number

√5=9-x^2÷4 is an irrational number

9-x^2 is an irrational number

x^2 is an irrational number

x is an irrational number

but we have assume that x is a rational number

we arrive at a contradiction

so our assumption that√5-2 is a rational number is wrong.

thus √5-2 is an irrational number