Solution - Radical equations

Radical Equation entered :

     √5-x-√x = 1

Step by step solution :

Step  1  :

Isolate a square root on the left hand side :

     Original equation
     √5-x-√x = 1

     Isolate
     √5-x = √x+1

Step  2  :

Eliminate the radical on the left hand side :

     Raise both sides to the second power
     (√5-x)² = (√x+1)²

     After squaring
     5-x = x+1+2√x

Step  3  :

Get remaining radical by itself :

     Current equation
     5-x = x+1+2√x

     Isolate radical on the left hand side
     -2√x = -5+x+x+1

      Tidy up
     2√x = 4-2x

Step  4  :

Eliminate the radical on the left hand side :

     Raise both sides to the second power
     (2√x)² = (4-2x)²

     After squaring
     4x = 4x²-16x+16

Step  5  :

Solve the quadratic equation :

     Rearranged equation
      4x²  - 20x  + 16 = 0

     This equation has two rational roots:
       {x1, x2}={4, 1}
 

Step  6  :

Check that the first solution is correct :

     Original equation, root isolated, after tidy up
     √5-x = √x+1

     Plug in 4 for  x 
      √5-(4) = √(4)+1

      Simplify
      √1 = 3
      Solution does not check
      1 ≠ 3 

Step  7  :

Check that the second solution is correct :

     Original equation, root isolated, after tidy up
     √5-x = √x+1

     Plug in 1 for  x 
      √5-(1) = √(1)+1

      Simplify
      √4 = 2
      Solution checks !!
     Solution is:
      x = 1

One solution was found :