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).

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 2. + 5 = 2 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 2. 6 3.  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) 2. (5 - i)(2 - 3i) 3. (4 - i)2 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 2.    3 - i

3 + i 3.    1 + 2i

3 - 5i 