prove that 3+2 root 5 is irrational
Answers
]Step-by-step explanation:
let us assume 3+2root 5 is in the form p/q where q not equal to 0 and p and q are coprimes.
3+2 root 5=p/q
5+root5=p/q
root 5=p/5q
LHS is irrational and RHS is rational
this is not possible
LHS not equal to RHS
hence our assumption is wrong
therefore 3+2root 5 is irrational
hence proved.
hey!
the answer to your first question is,
let 3+2root 5 be a rational number
3+2root5 = a/b ( a and b are co prime )
2root5 = a-3b/b
root5 = a-3b/2b
root5 and a-3b/2b are rational numbers
root5 = p/q ( p and q are co prime )
5 = p^2/q^2
5q^2 = p^2 .......... (1)
p^2 is divisible by 5
p is also divisible by 5
p = 5c ........... (2)
Put equation 2 in 1
5q^2 = 25c^2
q^2 = 5c^2
q^2 is divisible by 5
q is also divisible by 5
It contradict that p and q are co prime numbers. So, our supposition was wrong.
3+2root5 is an irrational number.
The answer to your second question is,
x^2 -2x-8 =0
p×q = -8
p+q = -2
So , p=-4 , q=2
x^2 +2x -4x -8 =0
x ( x + 2)-4 ( x +2) =0
( x-4)(x+2) =0
x = 4 , -2
Have a great day!