Sqrt(3x+1) - sqrt(x-1)=2?

Question

I have no idea how to do this, could someone explain it? Thank you.

Answer 1

sqrt(3x+1) - sqrt(x-1)=2

we have to find the domain first

{3x+1>=0

{x-1>=0

from the first one we obtain:

3x>=-1 ===> x>=-1/3

the second one:

x>=1

the domain is D:={x € IR such that x>=1}

now we can find the solution/s

first step:

sqrt(3x+1) - sqrt(x-1)=2 ===>

(sqrt(3x+1) - sqrt(x-1))² == 4.

second step:

(3x+1)+x-1 -2 sqrt( (3x+1)(x-1) )==4. then

4x-2 sqrt( 3x²-3x +x-1  )==4 ===>

-2 sqrt( 3x²-2x-1  ) == 4-4x

sqrt( 3x²-2x-1  ) == 4(1-x)/(-2)

sqrt( 3x²-2x-1  ) == 2x-2

step trhee

sqrt( 3x²-2x-1  )² == (2x-2)²

3x²-2x-1 == 4x²+4-8x

3x²-4x²-2x+8x-1-4 ==0

-x²+6x-5 ==0

x²-6x+5 ==0

now it is simple to see that x²-6x+5 can be factorized as :

x²-6x+5== (x-5)(x-1)

then the solutions are:

x1== 5 and x2== 1

proof:

for x== 1 we obtain:

sqrt(3+1) +0 == sqrt(4)==2 OK

for x== 5

sqrt(15+1) - sqrt(5-1)== sqrt(16)-sqrt(4)== 4-2 ==2 OK

Answer 2

It's simple:

If you square both sides up, you'll get:

3x+ 1 - x-1 = 4

which is, according to like times

3x-x + 1 -1 = 4

2x = 4

therefore x = 2

Answer 3

if you are trying to solve for X then this is how u do it.

sqrt(3x+1) - sqrt(x-1)=2

you square both sides

(3x+1) + (x-1) = 4

combine like terms

4x=4

divide by x

x=1

i used my calculator to double check and i got X=1 and X=5

Answer 4

Unfortunately, if you substitute Temitope's answer of x=2 back into the original equation, it doesn't work.  You can put both bracketed terms under one root sign, and then square it, getting (3x+1) - (x-1) = 4, which gives you 2x +2 = 4, and so 2x = 2 and x = 1.  If you substitute this back into the original equation, you get sqrt(3+1) - sqrt (1-1) = sqrt(4) = 2, it is correct.