Question

prove that 3-2root5 is irrational​

Answer 1

Answer:

Let us assume that 3-2 root 5 is rational

So, we will prove that Root 5 is rational

Root 5 = a/b ( b≠0, a and b are Co-primes)

Co- prime means that they can have only two factors 1 and the number itself)

Root 5 * b = a

Squaring both sides

5b² = a²

b² = a² / 5 ( Therefore 5 divides a² so 5 divides a) - (1)

Let a/c = 5

a = 5c

Squaring both sides

a² = 5c²

A² in above is 5b²

5b² = 25c²

( 5 and 25 gets cancelled in 5 table leaving 5 as quotient)

b² = 5c²

b²/5 = c² ( 5 divides b² so 5 divides b) - (2)

From 1 & 2

5 divides both a and b therefore violating our assumption that they are co-prime

therefore root 5 is irrational

hence our assumption false

2 is rational and root is irrational as proved above

Therefore rational * irrational = irrational

2 root 5 is irrational

3 is rational and 2 root 5 is irrational

Rational - irrational = irrational

Therefore 3-2 root 5 is irrational

Please mark as brainlist