Question

prove that (3+2√5)^2 is irrational.

Answer 1

Let take that 3 + 2√5 is a rational number.      So we can write this number as                                    3 + 2√5           = a/b      Here a and b are two co prime number and b is not equal to 0      Subtract 3 both sides we get                                    2√5                 = a/b – 3                                    2√5                 = (a-3b)/b      Now divide by 2 we get                                    √5                    = (a-3b)/2b      Here a and b are integer so (a-3b)/2b is a rational number so √5 should be a rational number But √5 is a irrational number so it contradict the fact                                       Hence result is 3 + 2√5 is a irrational number