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Chapter 3. Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations. Willa Cather –U.S. novelist.

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Chapter 3
Chapter 3

  • Complex Numbers

  • Quadratic Functions and Equations

  • Inequalities

  • Rational Equations

  • Radical Equations

  • Absolute Value Equations


Willa cather u s novelist
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.


Mathematics 116
Mathematics 116

  • Complex Numbers



Set of complex numbers
Set of Complex Numbers

  • R = real numbers

  • I = imaginary numbers

  • C = Complex numbers


Elbert hubbard
Elbert Hubbard

  • “Positive anything is better than negative nothing.”


Standard form of complex number
Standard Form of Complex number

  • a + bi

  • Where a and b are real numbers

  • 0 + bi = bi is a pure imaginary number


Equality of complex numbers
Equality of Complex numbers

  • a+bi = c + di

  • iff

  • a = c and b = d



Add and subtract complex s
Add and subtract complex #s

  • Add or subtract the real and imaginary parts of the numbers separately.


Orison swett marden
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.”


Multiply complex s
Multiply Complex #s

  • Multiply as if two polynomials and combine like terms as in the FOIL

  • Note i squared = -1


Complex conjugates
Complex Conjugates

  • a – bi is the conjugate of a + bi

  • The product is a rational number


Divide complex s
Divide Complex #s

  • Multiply numerator and denominator by complex conjugate of denominator.

  • Write answer in standard form


Harry truman american president
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.”


Calculator and complex s
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.


Mathematics 1161
Mathematics 116

  • Solving Quadratic Equations

  • Algebraically

  • This section contains much information


Def quadratic function
Def: Quadratic Function

  • General Form

  • a,b,c,are real numbers and a not equal 0


Objective solve quadratic equations
Objective – Solve quadratic equations

  • Two distinct solutions

  • One Solution – double root

  • Two complex solutions

  • Solve for exact and decimal approximations


Solving quadratic equation 1
Solving Quadratic Equation #1

  • Factoring

  • Use zero Factor Theorem

  • Set = to 0 and factor

  • Set each factor equal to zero

  • Solve

  • Check


Solving quadratic equation 2
Solving Quadratic Equation #2

  • Graphing

  • Solve for y

  • Graph and look for x intercepts

  • Can not give exact answers

  • Can not do complex roots.


Solving quadratic equations 3 square root property
Solving Quadratic Equations #3Square Root Property

  • For any real number c





Procedure
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




Dorothy broude
Dorothy Broude

  • “Act as if it were impossible to fail.”


Completing the square informal
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.


Procedure completing the square
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





Objective
Objective:

  • Solve quadratic equations using the technique of completing the square.


Mary kay ash
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.”


College algebra very important concept
College AlgebraVery Important Concept!!!

  • The

  • Quadratic

  • Formula


Objective of a students
Objective of “A” students

  • Derive

  • the

  • Quadratic Formula.


Quadratic formula
Quadratic Formula

  • For all a,b, and c that are real numbers and a is not equal to zero





Pearl s buck
Pearl S. Buck

  • “All things are possible until they are proved impossible and even the impossible may only be so, as of now.”


Methods for solving quadratic equations
Methods for solving quadratic equations.

  • 1. Factoring

  • 2. Square Root Principle

  • 3. Completing the Square

  • 4. Quadratic Formula


Discriminant
Discriminant

  • Negative – complex conjugates

  • Zero – one rational solution (double root)

  • Positive

    • Perfect square – 2 rational solutions

    • Not perfect square – 2 irrational solutions


Joseph de maistre 1753 1821 french philosopher
Joseph De Maistre (1753-1821 – French Philosopher

  • “It is one of man’s curious idiosyncrasies to create difficulties for the pleasure of resolving them.”




Calculator programs
CalculatorPrograms

  • ALGEBRAQUADRATIC

  • QUADB

  • ALG2

  • QUADRATIC


Ron jaworski
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.”


Objective1
Objective

  • Solve by Extracting Square Roots


Objective know and prove the quadratic formula
Objective: Know and Prove the Quadratic Formula

If a,b,c are real numbers and not equal to 0


Objective solve quadratic equations1
Objective – Solve quadratic equations

  • Two distinct solutions

  • One Solution – double root

  • Two complex solutions

  • Solve for exact and decimal approximations


Objective solve quadratic equations using calculator
Objective: Solve Quadratic Equations using Calculator

  • Graphically

  • Numerically

  • Programs

    • ALGEBRAA

    • QUADB

    • ALG2

    • others



D alembert french mathematician
D’Alembert – French Mathematician applied, real-life problems.

  • “The difficulties you meet will resolve themselves as you advance. Proceed, and light will dawn, and shine with increasing clearness on your path.”


Vertex
Vertex applied, real-life problems.

  • 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)


Axis of symmetry
Axis of symmetry applied, real-life problems.

  • The vertical line that goes through the vertex of the parabola.

  • Equation is x = constant


Objective2
Objective applied, real-life problems.

  • Graph, determine domain, range, y intercept, x intercept


Parabola with vertex h k
Parabola with vertex (h,k) applied, real-life problems.

  • Standard Form


Standard form of a quadratic function
Standard Form of a Quadratic Function applied, real-life problems.

  • 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


Find vertex
Find Vertex applied, real-life problems.

  • x coordinate is

  • y coordinate is


Vertex of quadratic function
Vertex of quadratic function applied, real-life problems.


Objective find minimum and maximum values of functions in real life applications
Objective: Find minimum and maximum values of functions in real life applications.

  • 1. Graphically

  • 2. Algebraically

    • Standard form

    • Use vertex

      3. Numerically


Roger maris new york yankees outfielder
Roger Maris, New York Yankees Outfielder real life applications.

  • “You hit home runs not by chance but by preparation.”


Objective3
Objective: real life applications.

  • Solve Rational Equations

    • Check for extraneous roots

    • Graphically and algebraically


Objective4
Objective real life applications.

  • Solve equations involving radicals

    • Solve Radical Equations

      Check for extraneous roots

    • Graphically and algebraically


Problem radical equation
Problem: radical equation real life applications.


Problem radical equation1
Problem: radical equation real life applications.


Problem radical equation2
Problem: radical equation real life applications.


Objective5
Objective: real life applications.

  • Solve Equations

  • Quadratic in Form


Objective6
Objective real life applications.

  • Solve equations

  • involving

  • Absolute Value


Procedure absolute value equations
Procedure:Absolute Value equations real life applications.

  • 1.Isolate the absolute value

  • 2. Set up two equations joined by “or”and so note

  • 3. Solve both equations

  • 4.Check solutions


Elbert hubbard1
Elbert Hubbard real life applications.

  • “Positive anything is better than negative nothing.”


Elbert hubbard2
Elbert Hubbard real life applications.

  • “Positive anything is better than negative nothing.”


Addition property of inequality
Addition Property of Inequality real life applications.

  • Addition of a constant

  • If a < b then a + c < b + c


Multiplication property of inequality
Multiplication property of inequality real life applications.

  • If a < b and c > 0, then ac > bc

  • If a < b and c < 0, then ac > bc


Objective7
Objective: real life applications.

  • Solve Inequalities Involving Absolute Value.

  • Remember < uses “AND”

  • Remember > uses “OR”

  • and/or need to be noted


Objective estimate solutions of inequalities graphically
Objective: Estimate solutions of inequalities graphically. real life applications.

  • Two Ways

    • Change inequality to = and set = to 0

    • Graph in 2-space

    • Or Use Test and Y= with appropriate window


Objective8
Objective: real life applications.

  • Solve Polynomial Inequalities

    • Graphically

    • Algebraically

    • (graphical is better the larger the degree)


Objectives
Objectives: real life applications.

  • Solve Rational Inequalities

    • Graphically

    • algebraically

  • Solve models with inequalities


Zig ziglar
Zig Ziglar real life applications.

  • “Positive thinking won’t let you do anything but it will let you do everything better than negative thinking will.”


Zig ziglar1
Zig Ziglar real life applications.

  • “Positive thinking won’t let you do anything but it will let you do everything better than negative thinking will.”


Mathematics 116 regression continued
Mathematics 116 Regression real life applications.Continued

  • Explore data: Quadratic Models and Scatter Plots


Objectives1
Objectives real life applications.

  • Construct Scatter Plots

    • By hand

    • With Calculator

  • Interpret correlation

    • Positive

    • Negative

    • No discernible correlation


Objectives2
Objectives: real life applications.

  • Use the calculator to determine quadratic models for data.

  • Graph quadratic model and scatter plot

  • Make predictions based on model


Napoleon hill
Napoleon Hill real life applications.

  • “There are no limitations to the mind except those we acknowledge.”


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