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Transcript
section 3.3

3.3 QUADRATIC EQUATIONS &
INEQUALITIES
QUIZ
Tell true or false of the following statement:
given linear functions f(x) and g(x), if (fg)(x) = 0,
then either f(x) = 0 or g(x) = 0.
QUADRATIC EQUATIONS
Solve quadratic equations by factoring
Solve by square root method
Solve by the quadratic formula.
QUADRATIC EQUATION IN ONE VARIABLE
Quadratic equation in standard form:
SOLVE BY FACTORING
Fact:
Given two functions f(x) and g(x), if (fg)(x) = 0, then either f(x) = 0
or g(x) = 0.
3y(y - 1) = 0
X2 – 49 = 0
X2 + 11x = 12
6p2 – 5p = 6
2x2 + 4x -16 = 0
X2 -6x + 9 =0
SQUARE ROOT METHOD
Fact:
Given a function f(x) , if ( f(x) )2 = B, B ≥ 0, then f(x) = ± √ B.
x2 = 49
(4x - 3)2 = 36
(1 - z)2 = 8
(a + 2)2 = -5
QUADRATIC FORMULA
The solutions of the quadratic equation ax2 + bx + c = 0,
where a ≠ 0, are given by the formula
x=
- b ± √ b2 – 4ac
x2 + 5x + 7 = 0
3p2 + 7p -2 = 0
(x - 3)(x + 5) = -7
2a
DISCRIMINANT
b2 – 4ac is called the discriminant of the
quadratic equation
Value of b2 – 4ac
Number of Solutions
Type of Solutions
Positive
Two
Real
0
One(identical solution)
Real
Negative
Two
Nonreal complex
DISCRIMINANT AND GRAPH
Value of b2 – 4ac
Number of Real
Solutions
No. of x-intercepts
Positive
Two
Two
0
One(identical solution)
One
Negative
None
None
SOLVING QUADRATIC INEQUALITIES
a>0
x1
x2
If x1 < x < x2, then f(x) < 0
If x = x1 or x2, then f(x) = 0
If x < x1 or x > x2, then f(x)> 0
a<0
x
x1
x2
If x1 < x < x2, then f(x) > 0
If x = x1 or x2, then f(x) = 0
If x < x1 or x > x2, then f(x)< 0
x
SOLVING QUADRATIC INEQUALITIES
1. Solve the corresponding quadratic equation
Finding the x- intercepts of the function
2. Identify the intervals determined by the
solutions of the equation
3. Use a test value from each interval to
determine which intervals from the solution set
SOLVING QUADRATIC INEQUALITIES
x2 + 4x -12 ≥ 0
x(x-7) < 8
x2 + 4x +4 < 0
x2 + 6x +9 > 0
2x2 + 4x +4 < 0
-x2 + 4x - 6 > 0
HOMEWORK
PG. 189: 5 – 50(M5),51,55,60
PG. 190: 63,66,68, 72 – 93(M3), 95 – 103(odds)
KEY: 30, 45, 51, 95
Reading: 3.4 Modeling
Reminder: Project 2
Skill test 2
Exam 2
Due March 2nd
Due March 4th
March 7th,8th