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Elementary Algebra Pqtz from Wikipedia, the Free Encyclopedia Contents Elementary algebra pqtz From Wikipedia, the free encyclopedia Contents 1 Parent function 1 1.1 See also ................................................ 1 1.2 External links ............................................. 1 2 Pointwise product 2 2.1 Formal definition ........................................... 2 2.2 Examples ............................................... 2 2.3 Algebraic application of pointwise products .............................. 3 2.4 Generalization ............................................. 3 2.5 See also ................................................ 3 3 Quadratic equation 4 3.1 Examples and applications ....................................... 4 3.2 Solving the quadratic equation ..................................... 5 3.2.1 Factoring by inspection .................................... 5 3.2.2 Completing the square .................................... 6 3.2.3 Quadratic formula and its derivation ............................. 7 3.2.4 Reduced quadratic equation .................................. 8 3.2.5 Discriminant ......................................... 8 3.2.6 Geometric interpretation ................................... 9 3.2.7 Quadratic factorization .................................... 9 3.2.8 Graphing for real roots .................................... 10 3.2.9 Avoiding loss of significance ................................. 11 3.3 History ................................................. 11 3.4 Advanced topics ............................................ 13 3.4.1 Alternative methods of root calculation ............................ 13 3.4.2 Generalization of quadratic equation ............................. 16 3.5 See also ................................................ 18 3.6 References ............................................... 18 3.7 External links ............................................. 20 4 Quadratic formula 21 4.1 Derivation of the formula ....................................... 21 i ii CONTENTS 4.2 Geometrical Significance ....................................... 22 4.3 Historical Development ........................................ 23 4.4 Other derivations ........................................... 23 4.4.1 Alternate method of completing the square .......................... 23 4.4.2 By substitution ........................................ 25 4.4.3 By using algebraic identities .................................. 25 4.4.4 By Lagrange resolvents .................................... 26 4.5 See also ................................................ 28 4.6 References ............................................... 28 4.7 External links ............................................. 29 5 Quartic function 30 5.1 History ................................................. 31 5.2 Examples ............................................... 31 5.3 Applications .............................................. 32 5.4 Inflection points and golden ratio ................................... 32 5.5 Solving a quartic equation ....................................... 32 5.5.1 Nature of the roots ...................................... 32 5.5.2 General formula for roots ................................... 33 5.5.3 Simpler cases ......................................... 34 5.5.4 Converting to a depressed quartic ............................... 36 5.5.5 Ferrari’s solution ....................................... 37 5.5.6 Solving by factoring into quadratics .............................. 38 5.5.7 Solving by Lagrange resolvent ................................ 39 5.5.8 Solving with algebraic geometry ............................... 40 5.6 See also ................................................ 40 5.7 References ............................................... 41 5.8 Further reading ............................................ 41 5.9 External links ............................................. 41 6 Rationalisation (mathematics) 42 6.1 Rationalisation of a monomial square root and cube root ....................... 42 6.2 Dealing with more square roots .................................... 43 6.3 Generalisations ............................................ 44 6.4 See also ................................................ 44 6.5 References ............................................... 44 7 Solving quadratic equations with continued fractions 45 7.1 A simple example ........................................... 45 7.2 An algebraic explanation ....................................... 46 7.3 The general quadratic equation .................................... 47 7.4 A general theorem ........................................... 47 CONTENTS iii 7.5 Complex coefficients ......................................... 48 7.6 See also ................................................ 49 7.7 References ............................................... 49 8 Transitive relation 50 8.1 Formal definition ........................................... 50 8.2 Examples ............................................... 50 8.3 Properties ............................................... 50 8.3.1 Closure properties ...................................... 51 8.3.2 Other properties ....................................... 51 8.3.3 Properties that require transitivity .............................. 51 8.4 Counting transitive relations ...................................... 51 8.5 See also ................................................ 51 8.6 Sources ................................................ 52 8.6.1 References .......................................... 52 8.6.2 Bibliography ......................................... 52 8.7 External links ............................................. 52 9 Trinomial 53 9.1 Trinomial expressions ......................................... 53 9.2 Trinomial equation .......................................... 53 9.3 See also ................................................ 53 9.4 References ............................................... 53 10 Two-element Boolean algebra 54 10.1 Definition ............................................... 54 10.2 Some basic identities ......................................... 54 10.3 Metatheory .............................................. 55 10.4 Footnotes ............................................... 56 10.5 See also ................................................ 56 10.6 References ............................................... 56 11 Unary operation 57 11.1 Unary negative and positive ...................................... 57 11.2 Examples from programming languages ................................ 57 11.2.1 C family of languages ..................................... 57 11.2.2 Unix Shell (Bash) ....................................... 58 11.2.3 Other languages ........................................ 58 11.3 See also ................................................ 58 11.4 References ............................................... 58 12 Unitary method 59 12.1 External links ............................................. 59 iv CONTENTS 13 Vieta’s formulas 60 13.1 The Laws ............................................... 60 13.1.1 Basic formulas ........................................ 60 13.1.2 Generalization to rings .................................... 61 13.2 Example ................................................ 61 13.3 Proof ................................................. 61 13.4 History ................................................ 61 13.5 See also ................................................ 62 13.6 References .............................................. 62 14 Zero-product property 63 14.1 Algebraic context ........................................... 63 14.2 Examples ............................................... 63 14.3 Non-examples ............................................. 64 14.4 Application to finding roots of polynomials .............................. 64 14.5 See also ................................................ 64 14.6 Notes ................................................. 65 14.7 References ............................................... 65 14.8 External links ............................................. 65 14.9 Text and image sources, contributors, and licenses .......................... 66 14.9.1 Text .............................................. 66 14.9.2 Images ............................................ 68 14.9.3 Content license ........................................ 69 Chapter 1 Parent function In mathematics, a parent function is the simplest function of a family of functions that preserves the definition (or shape) of the entire family. For example, for the family of quadratic functions having the general form y = ax2 + bx + c ; the simplest function is y = x2 This is therefore the parent function of the family of quadratic equations. For linear and quadratic functions, the graph of any function can be obtained from the graph of the parent function by simple translations and stretches parallel to the axes. For example, the graph of y = x2 − 4x + 7 can be obtained from the graph of y = x2 by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2)2. For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x). For example, the graph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle α along the positive X axis (where tan(α) = A⁄B), then stretching it parallel to the Y axis using a stretch factor R, where R2 = A2 + B2. This is because A sin(x) + B cos(x) can be written as R sin(x−α)
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