A New Look at Some Old Formulas

1985 ◽  
Vol 78 (1) ◽  
pp. 56-58
Author(s):  
Edward C. Wallace ◽  
Joseph Wiener
Keyword(s):  
Quadratic Formula ◽  
Quadratic Equations ◽  
Alternative Approaches ◽  
New Look

Interest in solving quadratic equations has occupied mathematicians for nearly four thousand years. Indeed, by 2000 b.c. the Babylonians had developed a form of the quadratic formula and a method equivalent to completing the square (Eves 1969; Smith 1951). A number of approaches to the solution of quadratic equations are possible. Let's examine some of these alternative approaches to see what new insights we might discover.

Computer-Oriented Mathematics: Quadratic Equations—Computer Style

1969 ◽  
Vol 62 (4) ◽  
pp. 305-309
Author(s):  
Walter Koetke ◽  
Thomas E. Kieren
Keyword(s):  
Quadratic Formula ◽  
Quadratic Equations ◽  
Modern Mathematics

ONE of the “old” topics that has been approached in a “new” way in modern mathematics is quadratic equations. No longer do students simply memorize the quadratic formula and do hundreds of exercises using it.

Sharing Teaching Ideas: A Squeeze Play on Quadratic Equations

1982 ◽  
Vol 75 (2) ◽  
pp. 132-136
Keyword(s):  
High School ◽  
Mathematics Teacher ◽  
Quadratic Equation ◽  
Senior Year ◽  
Quadratic Formula ◽  
Mathematics Courses ◽  
Quadratic Equations ◽  
Teaching Ideas

As a mathematics teacher whose present assignment is to teach science, I was somewhat dismayed when my physics class wa unable to solve a nontrivial quadratic equation. These students are all enrolled in senior-year mathematics and had taken all lower level mathematics courses available in our small Western Kansas high school. They charged this inability to having forgotten the quadratic formula. To the e students the quadratic formula is a magic passkey to solving “unfactorable” quadratic equations. On further di scussion, l discovered that they vaguely remembered having heard of the method of completing the square, but they saw no connection between the quadratic formula and that method of solving a quadratic equation. They could solve simple quadratics by hit-and-miss factoring, but that was their only tool with which to attack this problem.

Approaching quadratic equations from a right angle

2017 ◽  
Vol 101 (552) ◽  
pp. 424-438
Author(s):  
King-Shun Leung
Keyword(s):  
Secondary School ◽  
Mathematics Curriculum ◽  
Quadratic Equation ◽  
Bisection Method ◽  
Quadratic Formula ◽  
Secondary School Mathematics ◽  
Quadratic Equations ◽  
Newton Raphson ◽  
Raphson Method ◽  
Comprehensive Discussion

The theory of quadratic equations (with real coefficients) is an important topic in the secondary school mathematics curriculum. Usually students are taught to solve a quadratic equation ax2 + bx + c = 0 (a ≠ 0) algebraically (by factorisation, completing the square, quadratic formula), graphically (by plotting the graph of the quadratic polynomial y = ax2 + bx + c to find the x-intercepts, if any), and numerically (by the bisection method or Newton-Raphson method). Less well-known is that we can indeed solve a quadratic equation geometrically (by geometric construction tools such as a ruler and compasses, R&C for short). In this article we describe this approach. A more comprehensive discussion on geometric approaches to quadratic equations can be found in [1]. We have also gained much insight from [2] to develop our methods. The tool we use is a set square rather than the more common R&C. But the methods to be presented here can also be carried out with R&C. We choose a set square because it is more convenient (one tool is used instead of two).

4. Quadratic equations

2015 ◽  
pp. 40-52
Author(s):  
Peter M. Higgins
Keyword(s):  
General Equation ◽  
Solution Process ◽  
Quadratic Equation ◽  
Quadratic Formula ◽  
Quadratic Equations ◽  
Quadratic Approximations

A quadratic equation is one involving a squared term and takes on the form ax2 + bx + c = 0. Quadratic expressions are central to mathematics, and quadratic approximations are extremely useful in describing processes that are changing in direction from moment to moment. ‘Quadratic equations’ outlines the three-stage solution process. Firstly, the quadratic expression is factorized into two linear factors, allowing two solutions to be written down. Next is completing the square, which allows solution of any particular quadratic. Finally, completing the square is applied to the general equation to derive the quadratic formula that allows the three coefficients to be put into the associated expression, which then provides the solutions.

The Quadratic Formula—an Enrichment Approach

1972 ◽  
Vol 65 (5) ◽  
pp. 472-473
Author(s):  
Kenneth Stilwell
Keyword(s):  
Analytic Geometry ◽  
Quadratic Formula ◽  
Quadratic Equations ◽  
Simple Transformation

The procedure of completing the square, used in deriving the quadratic formula, is pervasive in mathematics. The following is presented as an alternate method of derivation of the quadratic formula for students studying analytic geometry. It is not intended to replace the traditional derivation, but rather is presented as an enrichment topic based on the student's ability to solve pure quadratic equations (that, is, equations of the form ax2 + b = 0, a ≠ 0) and perform a simple transformation.

Activities: A Graphical Approach to the Quadratic Formula

1996 ◽  
Vol 89 (1) ◽  
pp. 34-46
Keyword(s):  
Quadratic Functions ◽  
Quadratic Formula ◽  
Graphical Approach ◽  
Quadratic Equations

Introduction: Traditionally, the solution of quadratic equations has been taught before, and in isolation from, the study of quadratic functions. The quadratic formula itself has typically been derived by completing the square. Many teachers skip the derivation, and most students who see it do not fully understand it.

A Heuristic Method for Solving Polynomial Equations

1984 ◽  
Vol 77 (9) ◽  
pp. 710-714
Author(s):  
R D. Small
Keyword(s):  
Approximate Solution ◽  
Heuristic Method ◽  
Solution Process ◽  
Polynomial Equations ◽  
Numerical Techniques ◽  
Quadratic Formula ◽  
Quadratic Equations ◽  
Insight Into

“When you can see an approximate solution the battle is half over.” This statement is the theme of the following method for solving polynomial equations. When the quadratic formula generates solutions of quadratic equations, we gain little insight into the solution process. A little more insight can be gained by using numerical techniques such as those reviewed by Snover and Spikell (1979). since the convergence to the solution can be observed.

Social security and social welfare: a ‘new look’ from Hong Kong

1973 ◽  
Vol 2 (3) ◽  
pp. 225-238 ◽  
Author(s):  
T. S. Heppell
Keyword(s):  
Hong Kong ◽  
Social Security ◽  
Social Welfare ◽  
Recent Developments ◽  
Alternative Approaches ◽  
New Look

Many things are associated with Hong Kong, but innovation in social security and social welfare is not one of them. Yet recent developments in Hong Kong are perhaps of more than local interest alone, not simply because Hong Kong has the advantage of virtually starting from scratch in developing social security but also because it has, by force of circumstance as well by inclination, had to look for what have been called ‘Hong Kong solutions for Hong Kong problems’. The results of this process might be regarded, if not as models to be copied, at least as alternative approaches to be borne in mind when other countries review the possible choices of development in social security open to them.

Alternative approaches to ultra-high resolution imaging

1991 ◽  
Vol 49 ◽  
pp. 650-651
Author(s):  
J.M. Cowley
Keyword(s):  
High Resolution ◽  
High Voltage ◽  
High Resolution Electron Microscopy ◽  
Crystal Defects ◽  
High Resolution Imaging ◽  
General Applicability ◽  
Resolution Electron ◽  
Radiation Resistant ◽  
Resolution Imaging ◽  
Alternative Approaches

By extrapolation of past experience, it would seem that the future of ultra-high resolution electron microscopy rests with the advances of electron optical engineering that are improving the instrumental stability of high voltage microscopes to achieve the theoretical resolutions of 1Å or better at 1MeV or higher energies. While these high voltage instruments will undoubtedly produce valuable results on chosen specimens, their general applicability has been questioned on the basis of the excessive radiation damage effects which may significantly modify the detailed structures of crystal defects within even the most radiation resistant materials in a period of a few seconds. Other considerations such as those of cost and convenience of use add to the inducement to consider seriously the possibilities for alternative approaches to the achievement of comparable resolutions.

PEDIATRIC NEWS Unveils a New Look for the New Year

2010 ◽  
Vol 44 (1) ◽  
pp. 2
Author(s):  
Catherine Cooper Nellist ◽  
Mary Jo Dales
Keyword(s):  
New Look
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