Solve the equation below, show all steps please

Question

√x+1 - √x-2 =1

Answer 1

√(x + 1) - √(x - 2) = 1

√(x + 1)  = 1 + √(x - 2)         Square both sides

x + 1 = 1 + 2√(x - 2) + x - 2

2 = 2√(x - 2)

1 = √(x - 2)

1 = x - 2

4 = x

 

{4}

Answer 2

I'm guessing that you mean:

 

√(x+1) - √(x-2) = 1

 

First square both sides:

(x + 1) + (x - 2) - 2√((x+1)(x-2)) = 1.

 

(Remember that squaring may introduce extraneous solutions, so when you get the possible solutions, you'll want to check each one to eliminate any that don't work.)

 

Simplify:

2x - 1 - 2√((x+1)(x-2)) = 1

 

Get the radical alone on one side:

2x - 2 = 2√((x+1)(x-2))

 

Factor out 2:

x - 1 = √((x+1)(x-2))

 

Square both sides again and solve:

(x-1)² = (x+1)(x-2)

x² - 2x + 1 = x² - x - 2

-2x + 1 = -x - 2

3 = x

 

So x = 3.  Let's see if that works:

 

√(x+1) - √(x-2) = √(3+1) - √(3-2) = √4 - √1 = 2 - 1 = 1

 

Correct!