Concept of square root

Question

Number of solutions of equation,

!!sqrt(4x+4)- sqrt(4x-x)= sqrt(16x-16)!!

(a) 1

(b) 2

(c) 0

(d) infinite

Solution

!!sqrt(4x+4)- sqrt(4x-4)= sqrt(16x-16)!!

!!=> 2sqrt(x+1)-2sqrt(x-1)= 4 sqrt(x-1)!!

!!=> sqrt(x+1)- sqrt(x-1) = 2 sqrt(x-1)!!

!!=> sqrt(x+1) = 3 sqrt(x-1)!!

Squaring both the sides,

!!=> x+1 = 9(x-1)!!

!!=> 8x= 10 => x=5/4!!

To exist !!sqrt(x+1) and sqrt(x-1)!!

!!=> x+1 >=0 and x-1>=0!!

!!=> x>=-1 and x>=1!!

!!x!! should be greater than !!1!!.

So, one solution exist. Therefore option (a) is correct.