Solution - Radical equations

Radical Equation entered :

     √x = 7+√2

Step by step solution :

Step  1  :

Isolate a square root on the left hand side :

     Radical already isolated
     √x = 7+√2

Step  2  :

Eliminate the radical on the left hand side :

     Raise both sides to the second power
     (√x)² = (7+√2)²

     After squaring
     x = 2+49+14√2

Step  3  :

Get remaining radical by itself :

     Current equation
     x = 2+49+14√2

     Isolate radical on the left hand side
     -14√2 = -x+2+49

      Tidy up
     14√2 = x-51

Step  4  :

Eliminate the radical on the left hand side :

     Raise both sides to the second power
     (14√2)² = (x-51)²

     After squaring
     392 = x²-102x+2601

Step  5  :

Solve the quadratic equation :

     Rearranged equation
     x²  - 102x  + 2209 = 0

     This equation has two real roots:
       x1 = (102 + sqrt(1568) ) / 2 = 70.7990
      x2 = (102 - sqrt(1568) ) / 2 = 31.2010

Step  6  :

Check that the first solution is correct :

     Original equation, root isolated, after tidy up
     √x = 7+√2

     Plug in 70.7990 for  x 
      √(70.7990)  = 7+√2

      Simplify
      √70.799 = 8.414
      Solution checks !!
     Solution is:
      x = 70.7990

Step  7  :

Check that the second solution is correct :

     Original equation, root isolated, after tidy up
     √x = 7+√2

     Plug in 31.2010 for  x 
      √(31.2010)  = 7+√2

      Simplify
      √31.201 = 8.414
      Solution does not check
      5.586 ≠ 8.414 

One solution was found :