Algebra II
Chapter 5 Notes
PART II
5.5 Complex Numbers and Roots
Make a list of the first 15 perfect squares:
Simplify the radicals.
a.
36
b.
 4
c.
49
d.
 169
e.
64
Steps for simplifying radicals:
(1) Find the largest factor of the number that is a perfect square.
(2) Rewrite the expression as the __________________________ of _________ radical expressions
(3) Take the square root (of the __________________ ___________________) and leave the non-perfect
square in the radical symbol.
Together:
Simplify the radical.
28
 162
Simplify the radical expression.
1.
200
2.
44
3.
 20
4.
 80
5.
440
6.
 16
7.
 32
180
8.
i = _______ (which represents an imaginary unit)
Together:
Simplify the radical.
A.
 45
B.
  50
C.
Example 1:
a.
b.
c.
d.
e.
f.

1
 162
9
Steps for solving quadratic equations when there is no linear term (in the form ax² + c =0)
Step 1: Isolate ___________
Step 2: Take the _______________ of the square
(which is the ______________ _______________ to both sides of the equation)
Remember that you will have TWO solutions (one ___________, one ___________)
Example 2:
a.
b.
c.
d.
e.
f.
Example 3:
Solve.
1.
( x  3) 2  4  44
2.
3( x  3) 2  2  52
3.
( x  2) 2  5  45
4.
2( x  3) 2  2  58
Complex Conjugate:
Together (find the conjugate):
a. 3 – 5i
b. –5 + 3i
c. 25i – 17
Example 4:
a.
b.
c.
d.
e.
f.
d. –16i
5.6 The Quadratic Formula
The quadratic formula is used to solve quadratic functions in the form f(x) = ax² + bx + c
The formula is:
x
You must __________
the formula!!!!!!!
 b  b  4ac
2a
2
Example 1:
Find the zeros of the quadratic function by using the quadratic formula.
A.
f ( x)  2 x 2  16 x  27
B.
C.
f ( x)  2 x 2  5 x  3
D.
f ( x)  4 x 2  4 x  9
f ( x)  4 x 2  3 x  2
Find the zeros of the quadratic function by using the quadratic formula.
E.
f ( x)  9 x 2  6 x  11
F.
f ( x)  2 x 2  x  2
G.
f ( x)  2 x 2  x  15
H.
f ( x)  5 x 2  7 x  3
I.
f ( x)  x 2  3 x
J.
f ( x)  2 x 2  5
The discriminate is used to find the __________________ and __________________of
solutions for a quadratic function.
The formula for the discriminate is ______________________.
Discriminant
Quadratic Formula:
b 2  4ac  0
b 2  4ac  0
(_________ Number)
b 2  4ac  0
(_________)
(__________ Number)
y
y
y
_____ _________ solutions
x
x
x
____ _________ solution
____ __________ solutions
Example 3:
Find the type and number of solutions for each equation.
a.
x2  6x  7  0
d. x 2  4 x  4  0
b. x 2  6 x  9
e.
x 2  4 x  8
c.
x 2  11  6 x
f. x 2  2  4 x
5.7 Solving Quadratic Inequalities
Steps for graphing quadratic inequalities
1. Graph the parabola that defines the boundary
a. Use a _________________ line for ≥ and ≤
b. Use a _________________ line for > and <
2. Shade the appropriate region
a. Shade _________________________ the parabola for y > or ≥
b. Shade _________________________ the parabola for y ≤ or <
c. OR use a test point (typically 0,0) if true shade the region including the origin, if false
shade the region excluding the origin.
Example 1:
a.
b.
y
y
x
x
c.
d.
y
y
x
x
Example 2:
a.
b.
c.
d.
Example 3:
a.
b.
c.
d.
Example 4:
5.8
Pattern for quadratic function:____________________________________________
Example 1:
A.
B.
C.
D.
5.8 Calculator steps for finding a quadratic model for a given data set
[stat] [enter] enter data into L1 and L2
[stat]
[calc] [#5 QuadReg] [enter] [enter]
Record equation for the line of best fit (a is slope and b is y-intercept)
Record R²
Quadratic model: _________________________________________________________
Quadratic regression: _____________________________________________________
Example 2:
Example 3:
5.9
Complex numbers: _______________________________________________________
_______________________________________________________________________
Example 1:
A.
B.
C.
D.
E.
F.
G.
H.
i = _______
i² = ______
Example 2:
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
Example 3:
a.
b.
d.
e.
c.
Rationalize the denominator: _________________________________________________
________________________________________________________________________
________________________________________________________________________
Example 4:
a.
b.
c.
d.
e.
f.