Math 0311
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11.7 Complex Numbers
c) Multiply. 1  7i  3  4i 
Solution: Need to do Foil!!
1  7i  3  4i  
 1 3  4i   7i  3  4i 
Vocabulary/Terms/Stuff:
Imaginary numbers
Imaginary unit
Complex numbers
Conjugate of Complex Number
 3  4i  21i  28i 2
 3  17i  28 1
Thing to know how to do!
Identify Imaginary Numbers
Imaginary Number Calculations
1. Converting rational expression
2. Adding and Subtracting
3. Multiplying and Dividing
4. Powers of i
 3  28  17i
6  7i
d) Divide.
1  2i
Solution:
6  7i

1  2i
1. Perform the indicated operation and write in standard form.

 
a) Add or subtract. 8   8  5   50
Solution:
8   8  5   50 

 


6  12i  7i  14i 2 6  5i  14 1


1  4 1
1  2i  2i  4i 2
20  5i

 4i
5
 8  i 8  5  i 50
 13  2i 2  5i 2
 13  3i 2

 

 

Divide.

3  2i
1  8i
Solution:

 10  2i 6  4  5i 6
 10  2i 6  4  5i 6
 6  7i 6
6  7i  1  2i  6  7i 1  2i  61  2i   7i1  2i 



1  2i  1  2i  1  2i 1  2i 
1  2i  2i  4i 2


b) Add or subtract. 100   24  16   150
Solution:
100   24  16   150 
 31  17i

3  2i 3  2i  1  8i  3  24i  2i  16i 2 19  22i




1  8i 1  8i  1  8i 
65
1  64i 2
11
2. Find i
Solution: i1 , i 2  1, i 3  i, i 4  1, i 5  i, i 6  1,...
2R 3
4 11
Math 0311
So, i11  i 3  i .
12.1 SOLVING QUADRATIC EQUATIONS BY
COMPLETING THE SQUARE
Square Root Property:
If x2  a where a is a real number, then x   a .
Use the Square Root Property to Solve Equations
1. Solve the following equations.
a) x 2  9  0
Solution:
x2  9  0
x2  9
x2   9
x  3 or x  3
b) x  5  65
2
Solution:
x  5  65
2
x 2  60
2
3. Solve the equation.  a  4   32
Solution:
a  42  32
2
a  4   32
a  4  32
a  4  32 or a  4  32
Square of
a
Binominal
2
 x  4
 x  3
2
 x  5
2
Factors
Perfect Square
Trinominal
  x  4 x  4
 x 2  8 x  16
  x  3 x  3
 x2  6 x  9
  x  5 x  5
 x 2  10 x  25
Write Perfect Square Trinominals
A perfect square trinominal is a trinominal that can be expressed
as the square of a binominal.
x 2   60
x  2 15
Solve Quadratic Equations by Completing the Square
x  2 15 or x  2 15
Procedure: To Solve Quadratic Equations by Completing the
Square
1. Make leading coefficient equal to one. (multiply or divide
by a number)
2. Rewrite to have equation with only constant on the right.
3. Add to both sides: 1/2 the square of first-degree coefficient.
4. Factor trinominal to a binominal.
5. Use square root property.
6. Solve.
7. Check.
2. Solve the equation. x 2  7  0
Solution:
x2  7  0
x 2  7
x2    7
x  i 7 or x  i 7
Math 0311
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4. Solve the equation x 2  6 x  5  0 by completing the square.
Solution:
x 2  6x  5  0
x 2  6x 
 5
5. Solve the equation  x 2  3 x  18 by completing the square.
Solution:
 x 2  3 x  18
 x 2  3 x  18
x 2  6 x  9  5  9
x 2  3 x  18
 x  32  4
 x  32  
x 2  3x 
4
x  3  2
x  3  2
x  5 or x  1
 18
2
 3
 3
x  3x  
  18  

 2 
 2 
9
9
x 2  3 x   18 
4
4
2
3  184  9


x  
2
4
4

2
2
3
81

x  
2
4

2
3
81

x   
2
4

3
9
x 
2
2
3 9
x 
2 2
x  3 or x  6
2
Math 0311
6. Solve the equation by completing the square. x 2  2 x  48  0
Solution:
x 2  2 x  48  0
x 2  2 x  48
x 2  2 x  1  48  1
x  12  49
x  1  7
x  1  7
x  8 or x  6
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