Solution - Radical equations

Radical Equation entered :

     9 = 7√x+5√3-2x

Step by step solution :

Step  1  :

Isolate a square root on the left hand side :

     Original equation
     9 = 7√x+5√3-2x

     Isolate
     -7√x = -9+5√3-2x

      Tidy up
     7√x = 9-5√3-2x

Step  2  :

Eliminate the radical on the left hand side :

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

     After squaring
     49x = 75-50x+81-90√3-2x

Step  3  :

Get remaining radical by itself :

     Current equation
     49x = 75-50x+81-90√3-2x

     Isolate radical on the left hand side
     90√3-2x = -49x+75-50x+81

      Tidy up
     90√3-2x = -99x+156

Step  4  :

Eliminate the radical on the left hand side :

     Raise both sides to the second power
     (90√3-2x)² = (-99x+156)²

     After squaring
     24300-16200x = 9801x²-30888x+24336

Step  5  :

Solve the quadratic equation :

     Rearranged equation
      9801x²  - 14688x  + 36 = 0

     This equation has two real roots:
       x1 = (14688 + sqrt(214326000) ) /19602 = 1.4962
      x2 = (14688 - sqrt(214326000) ) /19602 = 0.0025

Step  6  :

Check that the first solution is correct :

     Original equation, root isolated, after tidy up
     7√x = 9-5√3-2x

     Plug in 1.4962 for  x 
      7√( 1.4962)  = 9-5√3-2•( 1.4962)

      Simplify
      7√1.496 = 8.562
      Solution checks !!
     Solution is:
      x =  1.4962

Step  7  :

Check that the second solution is correct :

     Original equation, root isolated, after tidy up
     7√x = 9-5√3-2x

     Plug in 0.0025 for  x 
      7√( 0.0025)  = 9-5√3-2•( 0.0025)

      Simplify
      7√0.002 = 0.347
      Solution checks !!
     Solution is:
      x =  0.0025

Two solutions were found :

1.       x =  0.0025
   
2.       x =  1.4962