Solution - Radical equations

Radical Equation entered :

     √2x+2+3x+5+√2x+2-3x+5 = 3

Step by step solution :

Step  1  :

Isolate a square root on the left hand side :

     Original equation
     √2x+2+3x+5+√2x+2-3x+5 = 3

     Isolate
     √2x+2+3x+5 = -√2x+2-3x+5+3

Step  2  :

Eliminate the radical on the left hand side :

     Raise both sides to the second power
     (√2x+2+3x+5)² = (-√2x+2-3x+5+3)²

     After squaring
     2x+2+3x+5 = 2x+2-3x+5+9-6√2x+2-3x+5

Step  3  :

Get remaining radical by itself :

     Current equation
     2x+2+3x+5 = 2x+2-3x+5+9-6√2x+2-3x+5

     Isolate radical on the left hand side
     6√2x+2-3x+5 = -2x-2-3x-5+2x+2-3x+5+9

      Tidy up
     6√2x+2-3x+5 = -6x+9

Step  4  :

Eliminate the radical on the left hand side :

     Raise both sides to the second power
     (6√2x+2-3x+5)² = (-6x+9)²

     After squaring
     72x+72-108x+180 = 36x²-108x+81

Step  5  :

Solve the quadratic equation :

     Rearranged equation
      36x²  - 72x  -171 = 0

     This equation has two real roots:
       x1 = ( 72 + sqrt(29808) ) / 72 = 3.3979
      x2 = ( 72 - sqrt(29808) ) / 72 = -1.3979

Step  6  :

Check that the first solution is correct :

     Original equation, root isolated, after tidy up
     √2x+2+3x+5 = -√2x+2-3x+5+3

     Plug in 3.3979 for  x 
      √2•( 3.3979) +2+3•( 3.3979) +5 = -√2•( 3.3979) +2-3•( 3.3979) +5+3

      Simplify
      √23.99 = 1.102
      Solution does not check
      4.898 ≠ 1.102 

Step  7  :

Check that the second solution is correct :

     Original equation, root isolated, after tidy up
     √2x+2+3x+5 = -√2x+2-3x+5+3

     Plug in -1.3979 for  x 
      √2•(-1.3979) +2+3•(-1.3979) +5 = -√2•(-1.3979) +2-3•(-1.3979) +5+3

      Simplify
      √0.01 = 0.102
      Solution checks !!
     Solution is:
      x = -1.3979

One solution was found :