Equations Reducible to Quadratic Form

Equations with radicals
The method:
Isolate a radical on one side of the equation, then square the resulting equation.
The squaring will get rid of the isolated radical. If another radical remains on the
other side of the equation, repeat the process.
The numbers obtained in this process are tentative, or extraneous solutions.
The numbers must be checked in the original radical equation. They may,
or may not work in the original radical equation

Case 1)   The equation contains only one radical.
Solve
      .
Isolate the radical,
      .
Square both sides of the equation.
    
     
Writing this in standard form,
     .
Factor the left side, is can use the Zero Product principle.

The possible solutions are
     
and
      .
These numbers must be checked in the original equation.
Check    :
     
           
                
This is a true statement, so    is a solution.

Check     :
     
            
               
This is a false statement, so    is not a solution.


Case 2)    The equation contains two radicals.
Use the method of  Case 1)  twice.
Solve
     .
First, isolate one of the radicals.
       
Square both sides of the equation.
     
Expand the right side by using the identity for the square of a difference.
        
         
Repeat the process of isolating the radical on one side of the equation.
       
This time, let's isolate the radical to the right side of the equation.
           
Now divide both sides of the equation by   because we can,
it simplifies the equation.
              
If the division resulted in a fraction, we probably would not
perform the division.
Square both sides of the equation.
        
           
            

Check:
     
             
                
This is a true statement, so is the solution.

See Example 1, pages 146 - 148.


Equations with Rational Exponents
Make an appropriate substitution in order to rewrite the equation in quadratic form.

Solve
     .

Using the law of exponents for raising a power, we can write

     
and

     .

The substitution    puts the equation in a form
that is more obviously quadratic,
        ,
which we can factor.
     
     
or
     
But recall that   ,  so
      corresponds to  ,
          corresponds to  .

To eliminate the fifth-roots in the two equations,
raise the equations to the fifth powers.
        or   

            or    .
The solution set is
              .

Equations with Higher Powers
Make an appropriate substitution in order to rewrite the equation in quadratic form.

Solve
     .
Using the law of exponents for raising a power, we can write
   .
The substitution
     
puts the equation in a form that is more obviously quadratic,
     .
Factor both sides of the equation.
     
          corresponds to 
whose solution is
         or   .

          corresponds to  
whose solution is
           or     .
The solution set is
                              .

See the example on page 148, and Examples 2 - 3, pages 149 - 150.


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