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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 readers 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

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Set of Complex Numbers R = real numbers I = imaginary numbers C = Complex numbers

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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

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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

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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

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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 #3 Square Root Property For any real number c

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Sample problem

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Sample problem 2

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Solve quadratics in the form

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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

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Sample problem 4

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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

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Sample Problem

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Sample Problem complete the square 2

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Sample problem complete the square #3

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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 shouldnt be able to fly, but the bumble bee doesnt know it so it goes flying anyway.

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College Algebra Very 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

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Sample problem quadratic formula #1

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Sample problem quadratic formula #2

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Sample problem quadratic formula #3

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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.

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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

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Joseph De Maistre (1753-1821 – French Philosopher It is one of mans curious idiosyncrasies to create difficulties for the pleasure of resolving them.

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Sum of Roots

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Product of Roots

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Calculator Programs ALGEBRA QUADRATIC QUADB ALG2 QUADRATIC

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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 Im going to complete every pass.

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Objective Solve by Extracting Square Roots

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Objective: Know and Prove the Quadratic Formula If a,b,c are real numbers and not equal to 0

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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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Objective: Solve Quadratic Equations using Calculator Graphically Numerically Programs –ALGEBRAA –QUADB –ALG2 –others

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Objective: Use quadratic equations to model and solve applied, real-life problems.

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DAlembert – 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.

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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

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

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Parabola with vertex (h,k) Standard Form

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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

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Find Vertex x coordinate is y coordinate is

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Vertex of quadratic function

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Objective: Find minimum and maximum values of functions in real life applications. 1. Graphically 2. Algebraically –Standard form –Use vertex 3. Numerically

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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

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Objective Solve equations involving radicals –Solve Radical Equations Check for extraneous roots –Graphically and algebraically

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Problem: radical equation

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Objective: Solve Equations Quadratic in Form

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Objective Solve equations involving Absolute Value

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Procedure:Absolute Value equations 1.Isolate the absolute value 2. Set up two equations joined by orand 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 0, then ac > bc If a 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 wont let you do anything but it will let you do everything better than negative thinking will.

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Zig Ziglar Positive thinking wont let you do anything but it will let you do everything better than negative thinking will.

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Mathematics 116 Regression Continued 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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