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## Chapter 3

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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**• “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**• Complex Numbers**Set of Complex Numbers**• R = real numbers • I = imaginary numbers • C = Complex numbers**Elbert Hubbard**• “Positive anything is better than negative nothing.”**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**• a+bi = c + di • iff • a = c and b = d**Add and subtract complex #s**• Add or subtract the real and imaginary parts of the numbers separately.**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 as if two polynomials and combine like terms as in the FOIL • Note i squared = -1**Complex Conjugates**• a – bi is the conjugate of a + bi • The product is a rational number**Divide Complex #s**• Multiply numerator and denominator by complex conjugate of denominator. • Write answer in standard form**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**• 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 116**• Solving Quadratic Equations • Algebraically • This section contains much information**Def: Quadratic Function**• General Form • a,b,c,are real numbers and a not equal 0**Objective – Solve quadratic equations**• Two distinct solutions • One Solution – double root • Two complex solutions • Solve for exact and decimal approximations**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**• Graphing • Solve for y • Graph and look for x intercepts • Can not give exact answers • Can not do complex roots.**Solving Quadratic Equations #3Square Root Property**• For any real number c**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**• “Act as if it were impossible to fail.”**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**• 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:**• Solve quadratic equations using the technique of completing the square.**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 AlgebraVery Important Concept!!!**• The • Quadratic • Formula**Objective of “A” students**• Derive • the • Quadratic Formula.**Quadratic Formula**• For all a,b, and c that are real numbers and a is not equal to zero**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.**• 1. Factoring • 2. Square Root Principle • 3. Completing the Square • 4. Quadratic Formula**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**• “It is one of man’s curious idiosyncrasies to create difficulties for the pleasure of resolving them.”**CalculatorPrograms**• ALGEBRAQUADRATIC • QUADB • ALG2 • QUADRATIC**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.”