Radical Equations and Complex Numbers

Radical Equations

If we have an equation with a single radical then we follow the procedure:

  • Step 1  Isolate the radical so that the radical is alone on the left side of the equation with everything else on the other side of the equation.

  • Step 2  Square both sides of the equation.

  • Step 3  Math 152A (old stuff).

  • Step 4  Check your answer for extraneous solutions.

 

Example

Solve 

        - 2 =  5


Solution

  1. =  7

  2. 7x + 4  =  (7)2  

    7x + 4  =  49

  3. 7x  =  45

    x  =  45/7

  4. Now plug in and verify:

    =  7 - 2  =  5    ok.

 

Exercises

Solve

 

  1. + 3 = 6        8/3

  2. + 5 = 2        No solution

Complex Numbers (Definitions)

Recall that we have defined the Natural, Whole, Integers, Rational, Irrational, and Real numbers.  We have also said that is not a real number.  

 

Definition of Complex Numbers

We define

           i =            (so that  i2 = -1)  

and let the Complex Numbers (C) to be the numbers of the form

          a + bi

where a and b are real numbers.  We call a the real part and b the imaginary part.  A complex number is called pure imaginary if a = 0.

 

 

Example

2+      =      2 +       =      2 + 3i

 

Exercise  

Put the following in complex form:

 

  1. + 8        8 +5i

  2. 6                     6 + 0i

  3.                 0 + 2 sqrt(3)i

Addition and Subtraction of Complex Numbers

Let a + bi and c + di be complex numbers, then

        (a + bi) + (c + di)  =  (a + c) + (b + d) i

 

Examples

        (2 - 3i) + (5 + 6i)  =  (2 + 5) + (-3 + 6) i  =  7 + 3i

        (4 + 2i) - (3 - i) = (4 - 3) + (2 + 1) i  =  1 + 3i

 

Multiplication of Complex Numbers

To multiply two complex numbers we jest use FOIL and remember that

         i2  =  -1 

 

Example

        (2 - 3i)(5 + i)  =  10 + 2i - 15i - 3i2  

        =  10 - 13i - 3(-1)  =  13 - 13i

 

Exercises

Multiply the complex numbers.

  1. (3 + 2i)(3 - 2i)        13

  2. (5 - i)(2 - 3i)          7 - 17i

  3. (4 - i)2                    15 - 8i

 

Division of Complex Numbers

Let a + bi be a complex number then we define the complex conjugate to be a - bi

We have 

        (a + bi) (a - bi)  =  a2 + b2

To divide complex numbers we multiply numerator and denominator by the complex conjugate.

 

Example

Divide

          5 - 3i
                         
          4 + 2i

 

Solution

Multiply top and bottom by 4 - 2i:

          (5 - 3i)(4 - 2i)
                                      
          (4 + 2i)(4 - 2i)

                 20 - 10i - 12i + 6i2
        =                                       
                         16 + 4

                 14 - 22i                7 - 11i
        =                         =                 
                    20                       10

                  7            11
        =               -            i
                 10           10

 

Exercises

Divide the following:

 

  1.    1
                          
       i

    -i

  2.    3 - i
                           
      3 + i

    4/5 - 3/5 i

  3.    1 + 2i
                          
       3 - 5i

    -7/34 + 11/34 i

 

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