Firstly know that:
sqrt(a)sqrt(b)=sqrt(ab)√a√b=√ab
And also that:
sqrt(q)sqrt(q)=q√q√q=q
Knowing this, let's find the value of x...
sqrt(x+3)-sqrt(x-1)=1√x+3−√x−1=1
(sqrt(x+3)-sqrt(x-1))^2=1^2(√x+3−√x−1)2=12
(sqrt(x+3)-sqrt(x-1))(sqrt(x+3)-sqrt(x-1))=1(√x+3−√x−1)(√x+3−√x−1)=1
x+3-sqrt(x+3)sqrt(x-1)-sqrt(x+3)sqrt(x-1)+(x-1)=1x+3−√x+3√x−1−√x+3√x−1+(x−1)=1
x+3-2sqrt(x+3)sqrt(x-1)+x-1=1x+3−2√x+3√x−1+x−1=1
2x+2-2sqrt(x+3)sqrt(x-1)=12x+2−2√x+3√x−1=1
2(x+1-sqrt(x+3)sqrt(x-1))=12(x+1−√x+3√x−1)=1
x+1-sqrt(x+3)sqrt(x-1)=1/2x+1−√x+3√x−1=12
x+1-1/2=sqrt((x+3)(x-1))x+1−12=√(x+3)(x−1)
x+1/2=sqrt((x+3)(x-1))x+12=√(x+3)(x−1)
(x+1/2)^2=(x+3)(x-1)(x+12)2=(x+3)(x−1)
x^2+1/2x+1/2x+1/4=x^2-x+3x-3x2+12x+12x+14=x2−x+3x−3
x^2+x+1/4=x^2+2x-3x2+x+14=x2+2x−3
x^2-x^2+1/4+3=2x-xx2−x2+14+3=2x−x
:. x=3+1/4