INSTRUCTORS Carleen Eaton Grant Fraser

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Dr. Carleen Eaton

Dr. Carleen Eaton

Quadratic Formula and the Discriminant

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Table of Contents

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
Section 2: Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
Section 3: Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
Section 4: Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
Section 5: Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
Section 6: Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
Section 7: Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
Section 8: Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
Section 9: Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
Section 10: Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
Section 11: Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (15)

0 answers

Post by William Zhang on May 1, 2020

Please answer my question

0 answers

Post by William Zhang on May 1, 2020

At 4:16 it should be -2 not -1

4 answers

Last reply by: Abigail Wu
Mon Jun 1, 2020 6:13 PM

Post by Jose Gonzalez-Gigato on December 5, 2011

At 4:15, in the section "One Rational Root", it should be -2, not -1 because it is -b. This will make the solution = -1, not - 1/2.

6 answers

Last reply by:
Thu Jan 23, 2020 9:00 PM

Post by Vasilios Sahinidis on December 25, 2010

I thought you set a=1 and b=2, in the quadratic formula you used 1 for b.

0 answers

Post by Jeffrey Petsko on September 25, 2010

it needs to discuss how to slove problems like 3x squared+1=2x!

Quadratic Formula and the Discriminant

  • The quadratic formula can be used to solve anyquadratic equation.
  • If you do not see an easy way to factor a quadratic equation, use the formula.
  • If the discriminant is positive, the equation has 2 real roots.
  • If the discriminant is 0, it has one rational root.
  • If the discriminant is negative, it has two complex conjugate roots.

Quadratic Formula and the Discriminant

Solve using the quadratic formula 3x2 + 5x = 12
  • Rewrite into standard form by subtracting 12 from both sides.
  • 3x2 + 5x − 12 = 0
  • Find a, b and c
  • a =
  • b =
  • c =
  • a = 3
  • b = 5
  • c = − 12
  • Plug the values into the quadratic formula x = [( − b ±√{b2 − 4ac} )/2a]
  • x = [( − 5 ±√{52 − 4(3)( − 12)} )/2(3)]
  • Simplify
  • x = [( − 5 ±√{52 − 4(3)( − 12)} )/2(3)] = [( − 5 ±√{25 + 144} )/6] = [( − 5 ±√{169} )/6]
  • Reduce square roots as much as possible
  • x = [( − 5 ±√{169} )/6] = [( − 5 ±13)/6]
x = [8/6] = [4/3] and x = − 3
Solve using the quadratic formula 2x2 − 72 = 7x
  • Rewrite into standard form ax2 + bx + c = 0
  • 2x2 − 7x − 72 = 0
  • Find a, b and c
  • a =
  • b =
  • c =
  • a = 2
  • b = − 7
  • c = − 72
  • Plug the values into the quadratic formula x = [( − b ±√{b2 − 4ac} )/2a]
  • x = [( − ( − 7) ±√{( − 7)2 − 4(2)( − 72)} )/2(2)]
  • Simplify
  • x = [(7 ±√{49 + 576} )/4] = [(7 ±√{625} )/4] = [(7 ±√{625} )/4]
  • Reduce square roots as much as possible
  • x = [(7 ±√{625} )/4] = [(7 ±25)/4]
x = [32/4] = 8 and x = − [18/4] = [9/2]
Solve using the quadratic formula 2x2 − 3 = 8x
  • Rewrite into standard form ax2 + bx + c = 0
  • 2x2 − 8x − 3 = 0
  • Find a, b and c
  • a =
  • b =
  • c =
  • a = 2
  • b = − 8
  • c = − 3
  • Plug the values into the quadratic formula x = [( − b ±√{b2 − 4ac} )/2a]
  • x = [( − ( − 8) ±√{( − 8)2 − 4(2)( − 3)} )/2(2)]
  • Simplify
  • x = [(8 ±√{64 + 24} )/4] = [(8 ±√{88} )/4] =
  • Reduce square roots as much as possible
  • x = [(8 ±√{88} )/4] = [(8 ±√{4*22} )/4] = [(8 ±√4 *√{22} )/4] = [(8 ±2√{22} )/4]
x = [(8 + 2√{22} )/4] = [(4 + √{22} )/2] and x = [(8 − 2√{22} )/4] = [(4 − √{22} )/2]
Solve using the quadratic formula 12x2 = 17 + 8x
  • Rewrite into standard form ax2 + bx + c = 0
  • 12x2 − 8x − 17 = 0
  • Find a, b and c
  • a =
  • b =
  • c =
  • a = 12
  • b = − 8
  • c = − 17
  • Plug the values into the quadratic formula x = [( − b ±√{b2 − 4ac} )/2a]
  • x = [( − ( − 8) ±√{( − 8)2 − 4(12)( − 17)} )/2(12)]
  • Simplify
  • x = [(8 ±√{64 + 816} )/24] = [(8 ±√{880} )/24] =
  • Reduce square roots as much as possible
  • x = [(8 ±√{880} )/24] = [(8 ±√{2*2*2*2*55} )/24] = [(8 ±4√{55} )/24] = [(8 ±4√{22} )/24] = [(4(2 ±1√{22} ))/4*6] = [(2 ±√{22} )/6]
x = [(2 + √{55} )/6] and x = [(2 − √{55} )/6]
Solve using the quadratic formula 3x2 − 11x + 8 = 0
  • Find a, b and c
  • a =
  • b =
  • c =
  • a = 3
  • b = − 11
  • c = 8
  • Plug the values into the quadratic formula x = [( − b ±√{b2 − 4ac} )/2a]
  • x = [( − ( − 11) ±√{( − 11)2 − 4(3)(8)} )/2(3)]
  • Simplify
  • x = [(11 ±√{121 − 96} )/6] = [(11 ±√{25} )/6] =
  • Reduce square roots as much as possible
  • x = [(11 ±√{25} )/6] = [(11 ±5)/6]
x = [(11 + 5)/6] = [16/6] = [8/3] and x = [(11 − 5)/6] = 1
Solve using the quadratic formula 3x2 − 12x − 96 = 0
  • Find a, b and c
  • a =
  • b =
  • c =
  • a = 3
  • b = − 12
  • c = − 96
  • Plug the values into the quadratic formula x = [( − b ±√{b2 − 4ac} )/2a]
  • x = [( − ( − 12) ±√{( − 12)2 − 4(3)( − 96)} )/2(3)]
  • Simplify
  • x = [(12 ±√{144 + 1152} )/6] = [(12 ±√{1296} )/6] =
  • Reduce square roots as much as possible
  • x = [(12 ±√{1296} )/6] = [(12 ±36)/6]
x = [(12 + 36)/6] = [48/6] = 8 and x = [(12 − 36)/6] = [( − 24)/6] = − 4
Solve using the quadratic formula 12x2 − 8x + 8 = 0
  • Find a, b and c
  • a =
  • b =
  • c =
  • a = 12
  • b = − 8
  • c = 8
  • Plug the values into the quadratic formula x = [( − b ±√{b2 − 4ac} )/2a]
  • x = [( − ( − 8) ±√{( − 8)2 − 4(12)(8)} )/2(12)]
  • Simplify
  • x = [(8 ±√{64 − 384} )/24] = [(8 ±√{ − 320} )/24] = [(8 ±i√{320} )/24]
  • Reduce square roots as much as possible
  • x = [(8 ±i√{320} )/24] = [(8 ±i√{2*2*2*2*2*2*5} )/24] = [(8 ±8i√5 )/24] = [(8(1 ±1i√5 ))/8*3] = [(1 ±i√5 )/3]
x = [(1 + i√5 )/3] and x = [(1 − i√5 )/3]
Solve using the quadratic formula 2x2 + 4x + 8 = 0
  • Find a, b and c
  • a =
  • b =
  • c =
  • a = 2
  • b = 4
  • c = 8
  • Plug the values into the quadratic formula x = [( − b ±√{b2 − 4ac} )/2a]
  • x = [( − (4) ±√{(4)2 − 4(2)(8)} )/2(2)]
  • Simplify
  • x = [( − 4 ±√{16 − 64} )/4] = [( − 4 ±√{ − 48} )/4] = [( − 4 ±i√{48} )/4]
  • Reduce square roots as much as possible
  • x = [( − 4 ±i√{48} )/4] = [( − 4 ±i√{2*2*2*2*3} )/4] = [( − 4 ±4i√3 )/4] = [(4( − 1 ±1i√3 ))/4] = − 1 ±i√3
x = − 1 − i√3 and x = − 1 + i√3
Find the discriminant and describe the nature of the solutions. − 3x2 − 3x − 12 = − 6
  • Write the quadratic in standard form.
  • − 3x2 − 3x − 6 = 0
  • Find a, b, c
  • a =
  • b =
  • c =
  • a = − 3
  • b = − 3
  • c = − 6
  • Calculate the discriminant D = b2 − 4ac
  • D = ( − 3)2 − 4( − 3)( − 6)
  • D = 9 − 72 = − 63
Since the discriminant is negative, there will be a pair of complex conjugates.
Find the discriminant and describe the nature of the solutions.
− 4x2 − 3x + 5 = 4
  • Write the quadratic in standard form.
  • − 4x2 − 3x + 1 = 0
  • Find a, b, c
  • a =
  • b =
  • c =
  • a = − 4
  • b = − 3
  • c = 1
  • Calculate the discriminant D = b2 − 4ac
  • D = ( − 3)2 − 4( − 4)(1)
  • D = 9 + 16 = 25
Since the discriminant is positive, there will be two real solutions.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Quadratic Formula and the Discriminant

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Quadratic Formula 0:21
    • Standard Form
    • Example: Quadratic Formula
  • One Rational Root 3:00
    • Example: One Root
  • Complex Solutions 6:16
    • Complex Conjugate
    • Example: Complex Solution
  • Discriminant 9:42
    • Positive Discriminant
    • Perfect Square (Rational)
    • Not Perfect Square (2 Irrational)
    • Negative Discriminant
    • Zero Discriminant
  • Example 1: Quadratic Formula 13:50
  • Example 2: Quadratic Formula 16:03
  • Example 3: Quadratic Formula 19:00
  • Example 4: Discriminant 21:33

Transcription: Quadratic Formula and the Discriminant

Welcome to Educator.com.0000

Today we are going to be discussing the quadratic formula.0002

In previous lessons, we talked about solving quadratic equations through methods such as graphing and completing the square.0005

However, those methods have certain limitations that the quadratic formula does not have.0014

Let's go ahead and take a look at this.0020

Now, notice that the solutions of this quadratic equation are given by this formula, the quadratic formula.0021

However, this specifies that the equation needs to be in standard form.0030

So, make sure you have your equation in standard form before you use the quadratic formula.0033

Given that it is in standard form, and has certain values of a, b, and c, this formula will give you the solutions for this equation.0039

Now, you need to make sure that you know this well; just memorize it.0048

And then, apply it to equations in standard form.0053

Now, an example would be if I was given 2x2 + x = 2.0057

This is not in standard form; so first, put it in standard form by subtracting 2 from both sides.0062

Now, I like to write out what a, b, and c are; that way, I can just plug them in without having to worry about making errors looking back at the equation.0082

So, here a equals 1; b equals 1; and c equals -2.0089

Now, using the quadratic equation, x = -b±√(b2 [1 squared] - 4a (which is 2) c (which is -2), divided by 2a (which is 2).0094

OK, this gives me x = -1 ±...1 squared is just 1, minus...4 times 2 is 8, times -2 gives me -16; 2 times 2 is 4.0119

Here, x equals -1, plus or minus...1 minus -16 is actually positive 17, over 4.0140

Therefore, x equals (-1 ±√17)/4.0151

You can leave it like this, or you could break it out into x = (-1 + √17)/4, or x = (-1 - √17)/4.0161

So, there are two solutions; and you found them by putting the equation in standard form, and then using the quadratic formula.0170

Now, in the example I just showed you, there were actually two roots, or two solutions, to that equation.0181

However, if b2 - 4ac equals 0, the equation has only one rational root.0187

So, let's take a look at the quadratic equation again, and figure out why that is so.0194

Let's apply this to an example, x2 + 2x + 1 = 0.0207

This is already in standard form, so I don't need to worry about changing it at all.0219

Let's take a look; and I have a = 1, b = 2, and c = 1.0226

Now, notice that this b2 - 4ac is just what is here under the radical sign.0232

And this is called the discriminant; and we will talk more about that in a minute.0240

But for right now, let's just look at it and realize that it is what is under the radical sign.0243

Let's actually try to solve this out: since a equals 1, I am going to get -1 ± the square root0252

of b2 (which is 22), minus 4 times a times c, over 2 times a.0260

OK, what I end up with here is 4 - 4, which is 0.0275

So, looking at this, when I plugged in my values, I got -1 plus or minus √(4 - 4), over 2; so that is -1 plus or minus √0, over 2.0292

Well, this would come out to -1 ± 0, over 2, which just equals -1/2.0306

There is only one solution--one rational root, or one rational solution.0314

And the reason is because, if b2 - 4ac = 0, this becomes 0.0321

And the reason you have two solutions is that you take this -b/2a, and then it is plus or minus this.0327

And you would get two solutions if, for example, b2 - 4ac is 4.0335

And if I had plus or minus 4, that is going to give me something plus 2, and then something minus 2--two different things.0341

However, if I am talking about plus or minus the square root of 0, I am just going to get 0.0355

I only have one number that I am going to come out with here.0360

Therefore, if this discriminant, b2 - 4ac, is 0, you are only going to get one root, or one solution, for your quadratic equation.0363

Complex solutions may occur; and if they do, the complex solutions will actually be a pair of complex conjugates.0377

Remember what a complex conjugate pair is, such as 2 + 3i and 2 - 3i; this would be an example of complex conjugates.0388

Let's think about how this can occur, looking at the quadratic formula.0402

I have the quadratic formula; and if this ends up being a negative number, you will end up with an imaginary number.0413

And so, then you will have a real part and an imaginary part.0425

And that is how you end up with solutions that are complex conjugates.0429

For example, if I was given 2x2 + 3x + 6 = 0, then I have a = 2, b = 3, and c = 6.0433

Applying the quadratic formula to this, I am going to get -3 ± √(b2,0447

which is 32, - 4 times 2 (which is a), times 6 (which is c)), all over 2 times a, which is 2.0461

So, x equals -3, plus or minus 9; and 4 times 2 is 8, times 6; that is 48; so 9 minus 48, all over 4.0474

This equals -3 plus or minus the square root of -39.0489

You can see, right away, that this is not going to give you a real number, because it is the square root of a negative number.0499

We can simplify this further by recalling that this equals this, and also recalling that the square root of -1 equals i.0506

This simplifies, therefore, to (i√39)/4.0523

OK, now I said that the solution is a pair of complex conjugates.0535

And if we would look at this carefully, I do have a set of complex conjugates.0539

because I have x = -3, plus i√39, over 4, and x = -3 minus i√39, all of this over 4.0543

Complex solutions occur when this under the radical ends up being negative.0563

So then, you end up with an imaginary part and a real part to this number.0570

And since we have plus or minus what is under here, it ends up being a pair of complex conjugates.0574

OK, I already mentioned the discriminant; and the discriminant is that expression under the square root sign.0582

The discriminant is very helpful, because it can tell you the nature of the solutions of the equation.0589

We already saw that; but looking a little more deeply into it, if the discriminant is positive, you get two solutions, and they are both real.0596

Looking back up here at the quadratic formula, x = (-b ± √(b2 - 4ac))/2a:0622

this right here is the discriminant; so if this is positive, the square root of a positive number is a real number; so both solutions will be real.0633

Now, we can break that down further--a subset.0644

We have that, if the discriminant is positive, we get two solutions, and they are both real.0648

If the discriminant is a perfect square, then the roots are rational.0652

And that makes sense, because let's say I end up with a perfect square:0670

if the discriminant here is a perfect square 4, then I am going to end up with 2.0675

And then, I will have -b plus or minus 2, over 2a; so that makes sense.0680

Now, if the discriminant is positive, but not a perfect square, then you get two irrational roots.0685

For example, if I figured out my discriminant, b2 - 4ac, and I ended up with something like 2,0716

(it is under the square root sign--that is my discriminant), this is not a perfect square;0723

well, that is an irrational number; so I am going to get -b ± √2, over 2a.0728

So, I am going to have two solutions, and they are real numbers, but they are irrational--two irrational roots.0735

OK, if the discriminant is negative, which we just saw, then there are no real solutions.0745

Instead, you can get a pair of complex conjugates, which is what we just saw.0771

Finally, if the discriminant is 0, then you get one real solution--one real root.0776

We saw that earlier today, as well, because if this comes out to be 0, you are going to get -b/2a ±0, which is just -b/2a; there is only one solution.0793

So, if the discriminant is positive, then you have two real solutions.0804

If there is a perfect square under here, the roots are rational.0808

If it is not a perfect square, then you can get two irrational roots.0811

If the discriminant is negative, there are no real solutions, but you can get a set of complex conjugates as solutions.0818

If the discriminant is 0, there is one real root.0824

OK, let's solve this using the quadratic formula, recalling that it is x = -b ± √(b2 - 4ac), all of that over 2a.0829

Here, a is 2; b is 3; and c is -4.0846

This gives me x = -3 ± √b2 (so that is 32) - 4(2)(-4), over 2(2).0852

So, let's see what this equals: -3 plus or minus...3 times 3 is 9, minus...4 times 2 is 8, times -4 is -32, over 2 times 2, which is 4.0879

This gives me 9 minus -32; let's just say + 32, over 4.0899

So, it is -3 ± √9 + 32, over 4, which equals -3 ± √...9 + 32 is 41, over 4.0907

OK, so what I have here are two roots; I have x = (-3 + √41)/4, and x = (-3 - √41)/4.0925

And what I ended up with are two solutions; they are both real, but they are irrational,0942

because what I have in the discriminant is not a perfect square.0950

Since this is not a perfect square (it is positive, so it is real, but it is not a perfect square), I end up with irrational roots.0956

OK, again, solve using the quadratic formula.0964

The first thing I have to see is that this is not in standard form.0966

So, I am going to put it into standard form by adding 18 to both sides.0970

Now, it is in standard form, but I can actually simplify this further, because I have a common factor over here (on this left side) of 2.0982

So, I am just going to divide both sides by 2 to make it simpler.0989

And that is going to give me x2 - 6x + 9 = 0.1002

OK, now I have a = 1, b = -6, and c = 9; so by dividing, I am working with smaller numbers, which is always easier.1009

Recall the quadratic formula: OK.1018

So, let's substitute in our values here: x equals -b, so that is -(-6), plus or minus the square root of -6 squared,1029

minus 4 times a times c, all over 2 times a.1043

A negative and a negative gives me a positive; plus or minus the square root of -6 squared, which is 36, minus... 4 times 1 is 4, times 9 is 36, over 2.1053

So, this equals 6, plus or minus the square root of...36 minus 36 is 0, over 2.1071

Now, you can probably already see that the discriminant is 0 here; so I am only going to get one real root as a solution,1086

because if I say 6 plus or minus the square root of 0 (that is 0) over 2, this just comes out to 6 over 2, which is 3.1093

So, the solution here is that x equals 3.1102

So, we started out with something that was not in standard form; I put this quadratic equation into standard form.1108

And then, to make it simpler, I divided both sides by this greatest common factor that was on the left, which is 2.1115

And then, I ended up with this: x2 - 6x + 9.1124

I used that to solve using the quadratic formula, and I quickly found out that this discriminant was 0.1128

So, I ended up with one real solution, which is x = 3.1136

All right, solve using the quadratic formula again.1143

x equals -b, plus or minus the square root of b2 minus 4ac, over 2a.1148

Now, a equals 1; b equals 1--it is already in standard form; and c equals 1.1157

So, these are nice, simple numbers that I am working with.1163

x equals -b, so that is -1, plus or minus the square root of b squared (that is 1 squared)1167

minus 4 times 1 times 1 again (minus 4ac), all over 2 times a, which is 1.1176

So, x equals -1 plus or minus the square root of 1 minus 4, all over 2.1187

So, x equals -1 plus or minus the square root of 1 - 4, which is -3.1196

OK, so you can see--what happened here is that I ended up with a negative number, a negative discriminant.1203

Because it is asking me to take the square root of a negative number, my solutions are going to be a pair of complex conjugates.1209

There are no real solutions, but there are solutions.1215

Recall that I can simplify this to say -1 plus or minus √-1, times √3, over 2.1220

And I am going to recall that i equals the square root of -1.1232

Therefore, x equals -1...I am going to change this into i...so I had the square root of -3;1237

I broke that down into the square root of -1, times the square root of 3, which gives me x = -1 ± i (since this is equal to i) √3, all over 2.1253

And I can rewrite that a slightly different way, and say x = -1 + i√3, over 2, and x = -1 - i√3, all over 2.1264

So, looking at this, what I have here are solutions that are complex conjugates.1279

Here, I am asked to find the discriminant and describe the nature of the solutions.1297

So, remember that the discriminant equals b2 - 4ac.1302

This is in standard form, so here I have a = 2, b = 3, and c = 7.1309

So, my discriminant is going to be 32 - 4(2)(7); so the discriminant equals 9...4 times 2 is 8, times 7 is 56; so the discriminant equals -47.1317

Here, the discriminant is negative, so the solutions will be a pair of complex conjugates,1335

just as we saw in the last problem, because the discriminant is negative, and you are going to have1349

to take the square root of that as part of the quadratic equation.1354

And that is going to end up giving you complex conjugates.1357

OK, that concludes this lesson of Educator.com on the quadratic formula.1361

And I will see you again next time.1366

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