Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical...
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Transcript of Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical...
![Page 1: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/1.jpg)
Chapter 3
• Complex Numbers
• Quadratic Functions and Equations
• Inequalities
• Rational Equations
• Radical Equations
• Absolute Value Equations
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Willa Cather –U.S. novelist
• “Art, it seems to me, should simplify. That indeed, is very nearly the whole of the higher artistic process; finding what conventions of form and what detail one can do without and yet preserve the spirit of the whole – so that all one has suppressed and cut away is there to the reader’s consciousness as much as if it were in type on the page.
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Mathematics 116
•Complex Numbers
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Imaginary unit i
2
1
1
i
i
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Set of Complex Numbers
• R = real numbers
• I = imaginary numbers
• C = Complex numbers
R I C
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Elbert Hubbard
–“Positive anything is better than negative nothing.”
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Standard Form of Complex number
• a + bi
• Where a and b are real numbers
• 0 + bi = bi is a pure imaginary number
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Equality of Complex numbers
• a+bi = c + di
• iff
• a = c and b = d
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Powers of i
1
2
3
4
1
1
i i
i
i i
i
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Add and subtract complex #s
• Add or subtract the real and imaginary parts of the numbers separately.
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Orison Swett Marden
• “All who have accomplished great things have had a great aim, have fixed their gaze on a goal which was high, one which sometimes seemed impossible.”
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Multiply Complex #s
• Multiply as if two polynomials and combine like terms as in the FOIL
• Note i squared = -1
2 1i
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Complex Conjugates
• a – bi is the conjugate of a + bi
• The product is a rational number
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Divide Complex #s
• Multiply numerator and denominator by complex conjugate of denominator.
• Write answer in standard form
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Harry Truman – American President
• “A pessimist is one who makes difficulties of his opportunities and an optimist is one who makes opportunities of his difficulties.”
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Calculator and Complex #s• Use Mode – Complex
• Use i second function of decimal point
• Use [Math][Frac] and place in standard form a + bi
• Can add, subtract, multiply, and divide complex numbers with calculator.
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Mathematics 116
• Solving Quadratic Equations
• Algebraically• This section contains much
information
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Def: Quadratic Function
• General Form
• a,b,c,are real numbers and a not equal 0
2( )f x ax bx c
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Objective – Solve quadratic equations
• Two distinct solutions
• One Solution – double root
• Two complex solutions
• Solve for exact and decimal approximations
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Solving Quadratic Equation #1
• Factoring• Use zero Factor Theorem• Set = to 0 and factor• Set each factor equal to zero• Solve• Check
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Solving Quadratic Equation #2
• Graphing
• Solve for y
• Graph and look for x intercepts
• Can not give exact answers
• Can not do complex roots.
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Solving Quadratic Equations #3Square Root Property
• For any real number c
2if x c then
x c or x c
x c
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Sample problem
2 40x 40x
4 10x
2 10x
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Sample problem 225 2 62x
25 60x 2 12x
12x 2 3x
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Solve quadratics in the form
2ax b c
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Procedure
• 1. Use LCD and remove fractions
• 2. Isolate the squared term
• 3. Use the square root property
• 4. Determine two roots
• 5. Simplify if needed
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Sample problem 3
23 16x
3 16x 3 4x
3 4 3 4 3 4x x or x
1 7 1, 7x or x
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Sample problem 4
27 25 2 3 0x
225 2 3 7x 2 7
2 325
x
7 72 3
25 5x i
3 71.5 0.26
2 10x i i
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Dorothy Broude
•“Act as if it were impossible to fail.”
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Completing the square informal
• Make one side of the equation a perfect square and the other side a constant.
• Then solve by methods previously used.
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Procedure: Completing the Square• 1. If necessary, divide so leading
coefficient of squared variable is 1.
• 2. Write equation in form
• 3. Complete the square by adding the square of half of the linear coefficient to both sides.
• 4. Use square root property
• 5. Simplify
2x bx k
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Sample Problem
2 8 5 0x x
4 11x
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Sample Problem complete the square 2
2 5 1 0x x 5 29
2x
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Sample problem complete the square #3
23 7 10 4x x
7 23
6 6x i
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Objective:
• Solve quadratic equations using the technique of completing the square.
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Mary Kay Ash
• “Aerodynamically, the bumble bee shouldn’t be able to fly, but the bumble bee doesn’t know it so it goes flying anyway.”
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College AlgebraVery Important Concept!!!
•The
•Quadratic
•Formula
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Objective of “A” students
• Derive
• the
• Quadratic Formula.
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Quadratic Formula
• For all a,b, and c that are real numbers and a is not equal to zero
23 8 7 0
4 5
3 3
x x
x i
2 4
2
b b acx
a
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Sample problem quadratic formula #1
22 9 5 0x x 1
, 52
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Sample problem quadratic formula #2
2 12 4 0x x
6 2 10x
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Sample problem quadratic formula #3
23 8 7 0x x 4 5
3 3x i
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Pearl S. Buck
• “All things are possible until they are proved impossible and even the impossible may only be so, as of now.”
![Page 44: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/44.jpg)
Methods for solving quadratic equations.
• 1. Factoring
• 2. Square Root Principle
• 3. Completing the Square
• 4. Quadratic Formula
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Discriminant
• Negative – complex conjugates• Zero – one rational solution (double
root)• Positive
– Perfect square – 2 rational solutions– Not perfect square – 2 irrational
solutions
2 4b ac
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Joseph De Maistre (1753-1821 – French Philosopher
• “It is one of man’s curious idiosyncrasies to create difficulties for the pleasure of resolving them.”
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Sum of Roots
1 2
br r
a
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Product of Roots
1 2
cr r
a
![Page 49: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/49.jpg)
CalculatorPrograms
• ALGEBRAQUADRATIC
• QUADB
• ALG2
• QUADRATIC
![Page 50: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/50.jpg)
Ron Jaworski
• “Positive thinking is the key to success in business, education, pro football, anything that you can mention. I go out there thinking that I’m going to complete every pass.”
![Page 51: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/51.jpg)
Objective
• Solve by Extracting Square Roots
2 0If a c where c
then a c
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Objective: Know and Prove the Quadratic Formula
If a,b,c are real numbers and not equal to 0
2 4
2
b b acx
a
![Page 53: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/53.jpg)
Objective – Solve quadratic equations
• Two distinct solutions
• One Solution – double root
• Two complex solutions
• Solve for exact and decimal approximations
![Page 54: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/54.jpg)
Objective: Solve Quadratic Equations using Calculator
• Graphically• Numerically• Programs
– ALGEBRAA– QUADB– ALG2– others
![Page 55: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/55.jpg)
Objective: Use quadratic equations to model and solve applied, real-life problems.
![Page 56: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/56.jpg)
D’Alembert – French Mathematician–“The difficulties you meet will
resolve themselves as you advance. Proceed, and light will dawn, and shine with increasing clearness on your path.”
![Page 57: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/57.jpg)
Vertex
• The point on a parabola that represents the absolute minimum or absolute maximum – otherwise known as the turning point.
• y coordinate determines the range.
• (x,y)
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Axis of symmetry
• The vertical line that goes through the vertex of the parabola.
• Equation is x = constant
![Page 59: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/59.jpg)
Objective
• Graph, determine domain, range, y intercept, x intercept
2
2
y x
y ax
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Parabola with vertex (h,k)
• Standard Form
2y a x h k
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Standard Form of a Quadratic Function
• Graph is a parabola
• Axis is the vertical line x = h
• Vertex is (h,k)
• a>0 graph opens upward
• a<0 graph opens downward
2( ) ( )f x a x h k
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Find Vertex
• x coordinate is
• y coordinate is
2
b
a
2
bf
a
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Vertex of quadratic function
,2 2
b bf
a a
![Page 64: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/64.jpg)
Objective: Find minimum and maximum values of functions in real
life applications.
• 1. Graphically
• 2. Algebraically
–Standard form
–Use vertex
3. Numerically
![Page 65: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/65.jpg)
Roger Maris, New York Yankees Outfielder
•“You hit home runs not by chance but by preparation.”
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Objective:
• Solve Rational Equations
–Check for extraneous roots
–Graphically and algebraically
![Page 67: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/67.jpg)
Objective
• Solve equations involving radicals
–Solve Radical Equations
Check for extraneous roots
–Graphically and algebraically
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Problem: radical equation
3 2 4 2 0x
6
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Problem: radical equation
1 7x x
10
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Problem: radical equation
2 3 2 2x x
23
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Objective:
• Solve Equations
• Quadratic in Form
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Objective
• Solve equations
• involving
• Absolute Value
![Page 73: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/73.jpg)
Procedure:Absolute Value equations
• 1.Isolate the absolute value• 2. Set up two equations joined
by “or”and so note• 3. Solve both equations• 4.Check solutions
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Elbert Hubbard
• “Positive anything is better than negative nothing.”
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Elbert Hubbard
• “Positive anything is better than negative nothing.”
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Addition Property of Inequality
• Addition of a constant
• If a < b then a + c < b + c
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Multiplication property of inequality
• If a < b and c > 0, then ac > bc
• If a < b and c < 0, then ac > bc
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Objective:
• Solve Inequalities Involving Absolute Value.
• Remember < uses “AND”
• Remember > uses “OR”
• and/or need to be noted
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Objective: Estimate solutions of inequalities graphically.
• Two Ways– Change inequality to = and set = to 0– Graph in 2-space
– Or Use Test and Y= with appropriate window
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Objective:
• Solve Polynomial Inequalities
–Graphically
–Algebraically
–(graphical is better the larger the degree)
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Objectives:
• Solve Rational Inequalities
–Graphically
–algebraically
• Solve models with inequalities
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Zig Ziglar
• “Positive thinking won’t let you do anything but it will let you do everything better than negative thinking will.”
![Page 83: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/83.jpg)
Zig Ziglar
• “Positive thinking won’t let you do anything but it will let you do everything better than negative thinking will.”
![Page 84: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations.](https://reader035.fdocuments.in/reader035/viewer/2022081422/5519cecf550346443e8b49a5/html5/thumbnails/84.jpg)
Mathematics 116 RegressionContinued
• Explore data: Quadratic Models and Scatter Plots
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Objectives• Construct Scatter Plots
– By hand
– With Calculator
• Interpret correlation
– Positive
– Negative
– No discernible correlation
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Objectives:
• Use the calculator to determine quadratic models for data.
• Graph quadratic model and scatter plot
• Make predictions based on model
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Napoleon Hill
• “There are no limitations to the mind except those we acknowledge.”
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