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The Honeybee Conjecture

noreply@blogger.com (Abdulla Nafiz) — Wed, 15 May 2013 17:24:00 +0000

Solved! A bee-buzzing, honey-licking 2,000-year-old mystery that begins here, with this beehive. Look at the honeycomb in the photo and ask yourself: (I know you've been wondering this all your life, but have been too shy to ask out loud ... ) Why is every cell in this honeycomb a hexagon?

Picture courtesy http://www.quick-good-fortune.com

Read this interesting article in full at Krulwich Wonders

The Accidents of Youth (1819)

noreply@blogger.com (Abdulla Nafiz) — Wed, 17 Apr 2013 13:21:00 +0000

Through a series of short stories and wonderful engravings, this book is aimed at keeping young people out of trouble and “calculated to improve [their] moral conduct”. As the author declares in his/her brilliantly earnest preface addressed to the child reader of the book:

My Dear Children, The inexperience and thoughtlessness natural at your age exposes you to many dangers : I have therefore pointed out some of them in this book, which contains several instructive little histories, in which you will behold the misfortunes that arise from disobedience and want of thought. When your parents desire you not to climb upon the chairs, or touch the fire, or play with knives, or pins, it is not because they wish to prevent you amusing yourselves ; they are only anxious to keep you from harm. If you were allowed to do whatever you pleased, many accidents would happen through your own indiscretion : for instance, when climbing on the furniture you might fall, and break a leg or an arm; and might burn yourselves, by playing with fire ; or cut your fingers, by playing with knives ; or might swallow pins, in putting them into your mouth. Thus, you see you might often lame or kill yourselves, if your good mamma or papa did not guard most of your actions. Do not suppose, my dear children, that I wish to prevent your playing and taking proper exercise. On the contrary, I am very much amused by your games, though they are sometimes noisy; and I admire your harmless mirth. I wish you to be gay and to amuse yourselves at proper times ; but you should never be rash or disobedient. You can play very well without climbing up to the window, on the furniture, or other improper places. If you see guns, pistols, or other dangerous weapons, you should never touch them, as you may always find play-things more agreeable, and free from danger. Why should you play with a knife or with the fire? and why put things into your mouth, at the risk of poisoning yourselves? These things are forbidden you; and yet your amusement is the wish of your parents. They desire only to see you happy, and guard you against accidents which your own discretion would not avoid. If you are good children, you will pay attention to the advice of your friends, and receive it as a proof of their love.

The book is housed at the Internet Archive, contributed by the California Digital Library.

Why Leaders Succeed and Bosses Fail

noreply@blogger.com (Abdulla Nafiz) — Sat, 06 Apr 2013 18:43:00 +0000

We all know that every leader is a boss, but not every boss is a leader. We can break down the fundamental differences between a boss and a true leader in terms of opposites. Do you talk, or listen? Do you demand, or motivate? If you are a boss, you are followed because of authority. If you are a leader, you are followed because of respect and admiration.

Exceptional leadership begins with a positive view of people and ends with remarkable accomplishments. Consider the differences between a boss and a leader in the following areas.

A boss demands, a leader consults.

A boss says, in a very loud and demanding voice, “I need you to get this done now!” In most cases, such a command is probably met with disdain and frustration. Did this approach cause the employees to deliver a sloppy and sub-par an outcome? It most likely did. We should all be wary of barking out orders to workers who hope to be treated with respect and kindness. When a boss demands results and places stringent expectations upon an employee without asking for perspective, his or her credibility suffers.

Read full article @ Forbes

I'm sorry, but I don't want to be an emperor

noreply@blogger.com (Abdulla Nafiz) — Fri, 15 Mar 2013 19:41:00 +0000

“I'm sorry, but I don't want to be an emperor. That's not my business. I don't want to rule or conquer anyone. I should like to help everyone if possible; Jew, Gentile, black man, white. We all want to help one another.

Human beings are like that. We want to live by each other's happiness, not by each other's misery. We don't want to hate and despise one another. In this world there is room for everyone, and the good earth is rich and can provide for everyone. The way of life can be free and beautiful, but we have lost the way.

Greed has poisoned men's souls, has barricaded the world with hate, has goose-stepped us into misery and bloodshed. We have developed speed, but we have shut ourselves in. Machinery that gives abundance has left us in want. Our knowledge has made us cynical; our cleverness, hard and unkind. We think too much and feel too little. More than machinery, we need humanity. More than cleverness, we need kindness and gentleness. Without these qualities, life will be violent and all will be lost.

The airplane and the radio have brought us closer together. The very nature of these inventions cries out for the goodness in men; cries out for universal brotherhood; for the unity of us all. Even now my voice is reaching millions throughout the world, millions of despairing men, women, and little children, victims of a system that makes men torture and imprison innocent people. To those who can hear me, I say, do not despair. The misery that is now upon us is but the passing of greed, the bitterness of men who fear the way of human progress. The hate of men will pass, and dictators die, and the power they took from the people will return to the people. And so long as men die, liberty will never perish.

Soldiers! Don't give yourselves to brutes, men who despise you, enslave you; who regiment your lives, tell you what to do, what to think and what to feel! Who drill you, diet you, treat you like cattle, use you as cannon fodder. Don't give yourselves to these unnatural men - machine men with machine minds and machine hearts! You are not machines, you are not cattle, you are men! You have the love of humanity in your hearts! You don't hate! Only the unloved hate; the unloved and the unnatural. Soldiers! Don't fight for slavery! Fight for liberty!

(this portion is censored..sorry!) You, the people, have the power, the power to create machines, the power to create happiness! You, the people, have the power to make this life free and beautiful, to make this life a wonderful adventure. Then in the name of democracy, let us use that power. Let us all unite. Let us fight for a new world, a decent world that will give men a chance to work, that will give youth a future and old age a security.

By the promise of these things, brutes have risen to power. But they lie! They do not fulfill that promise. They never will! Dictators free themselves but they enslave the people. Now let us fight to fulfill that promise. Let us fight to free the world! To do away with national barriers! To do away with greed, with hate and intolerance! Let us fight for a world of reason, a world where science and progress will lead to all men's happiness. Soldiers, in the name of democracy, let us all unite!”
― Charles Chaplin

Lecture on Income Inequality

noreply@blogger.com (Abdulla Nafiz) — Sat, 02 Mar 2013 16:10:00 +0000

Lecture by Prof. Emmanuel Saez.

College Degree Required. But Why?

noreply@blogger.com (Abdulla Nafiz) — Fri, 22 Feb 2013 20:32:00 +0000

A bachelor's degree is now required for jobs that used to be held by high school grads. Could this actually hurt your business?

There's an idea that the person with a degree is "better" than a person without one. Indeed, The New York Times recently reported on the relatively new phenomenon of companies hiring people with college degrees for jobs that historically didn't require college degrees. Are you doing this in your business?

If so, I have to ask, better for what? Yes, having a four year degree does show a degree of dedication. You have to pick a major, take class after class, write paper after paper and work on dreaded group projects. (Which, in my humble opinion, should be banished off the face of the educational earth unless the professor is willing to act as a proper manager, which most are not.) But, anyway, in theory you learn some things and you demonstrate that you have stick-to-itiveness. This is worth something.

But what? You also have to assume that no one enrolled in college, shelled out fantastic amounts of tuition and studied for hours to memorize the philosophies of 40 different dead people with ambitions of becoming an administrative assistant. No, they had other goals.

Visit Inc. to read full article by Suzanne Lucas..

Homework for nafix.com

noreply@blogger.com (Abdulla Nafiz) — Mon, 22 Oct 2012 12:15:00 +0000

Hi there!

Don't worry, this isn't really homework. I was checking out nafix.com and I saw your page titled "Completing the square." Here's the link in case you don't remember it.

www.nafix.com/2011/09/completing-square-and-quadratic.html

A lot of teachers and students aren't aware of the free teaching resource videos and homework help tutorials that are available online. There is an awesome site called HippoCampus.org that covers all the subjects--and it's supported by the Bill and Melinda Gates Foundation.

Check out some of their videos on algebra:

Here's the link.

I was hoping that if you liked it, you wouldn't mind sharing it with your readers by mentioning it on your site. If math isn't your subject, HippoCampus.org also offers free help for science, history, government and more so maybe you would just like to recommend HippoCampus.org.

Thanks!

Mike Joba
425 North Prince Street
Lancaster, PA 17601

The Dark Knight's Dark Secret

noreply@blogger.com (Abdulla Nafiz) — Mon, 15 Oct 2012 21:27:00 +0000

During the closing credits of the summer's blockbuster film, The Dark Knight Rises, the screen displays the following title: "Based upon Batman characters created by Bob Kane." Similar declarations appear on every Batman film, TV show, video game and comic book. These statements are, at best, only half truths.

Batman was created by two men. One, Bob Kane, found wealth and fame in his creation and has subsequently been identified as the character's sole creator. The other, Bill Finger, has never received official credit for his role in the character's origin or shared in the wealth generated by the licensing it has produced.

Author Marc Tyler Nobleman hopes to change that. His recent picture book, Bill: The Boy Wonder: The Secret Co-Creator of Batman, is the first to focus on Finger's life story. Nobleman had previously written a similar illustrated book on Superman creators Jerry Siegel and Joe Shuster, and then turned his attention to the next great comic book superhero: Batman.

Following the runaway success of Superman in 1938, DC Comics (then known as National Periodical Publications) was looking for a follow-up superhero title. On a Friday in late 1938, DC editor Vin Sullivan asked artist Kane for help. Over the weekend, Kane got together with his friend, writer Finger, to work through ideas.

The details of who created which aspects of Batman's characterization remain murky, with Kane telling different versions at different times, and Finger only mentioning his involvement late in his life. Most accounts credit Kane with the original notion of a bird-like or bat-like character with wings and a red suit. Nobleman contends, however, that most of the central elements of Batman's mythos can be attributed to Finger: the gray and black costume, the bat-like cowl, the batmobile and many of the colorful villains that Batman faced. More significantly, Finger wrote the early Batman stories and created the tale of the character's tragic origin.

Finger received no credit as a writer in the original Batman comic books, a fairly common practice at the time. In addition to relying on Finger's scripting, Kane also used uncredited ghost artists. Although the details of the arrangement are a matter of speculation, at some point -- perhaps as early as 1946 -- Kane struck a deal with DC Comics to be identified as the creator of Batman.

The launch of the "Batman" television program in 1966 made Kane a great deal of money and turned him into something of a small-scale celebrity. Finger scripted one episode of the show, but otherwise saw none of the reward from the character's growing popularity.

Finger's role in the creation of Batman was finally brought to light during his appearance at an early comic book convention in New York in 1965. Later that year, comic book historian Jerry Bails published an article titled, "If the Truth Be Known or 'A Finger in Every Plot!'" in a relatively obscure fanzine that described Finger's contributions to Batman's origin. Kane wrote a response in another fan magazine contradicting Finger's claims, stating "I, Bob Kane, am the sole creator of 'Batman.'"

Finger died in 1974, a few weeks before his 60^th birthday, with his work on Batman still officially unrecognized and little known outside the insular world of comic book fandom. Kane passed away in 1998 at age 83.

When Nobleman set out to write an illustrated book about Finger's life, he sought photo references to allow the book's illustrator, Ty Templeton, to bring Finger to life. Industry insiders told Nobleman only two photos of Finger existed. He was also told that Finger had no living heir to receive any royalties from Finger's creations. During his research for the book, Nobleman discovered that neither of these points was true.

Knowledge@Wharton sat down with Nobleman to discuss Finger's life and legacy -- and what lessons his story has for creative artists today. An edited version of that conversation follows.

Read full article here

The Power of Appreciation

noreply@blogger.com (Abdulla Nafiz) — Mon, 08 Oct 2012 20:03:00 +0000

Building Social and Emotional Skills in Elementary Students: The Power of Appreciation

In this nine-part series, we will look at important factors that influence the happiness and social and emotional learning of elementary school age children. These factors are very useful in helping students learn, manage emotions and increase empathy. Each blog features one letter of the acronym HAPPINESS:

H = Happiness
A = Appreciation
P = Passions and Strengths
P = Perspective
I = Inner Meanie, Inner Friend
N = Ninja Mastery
E = Empathy
S = So Similar
S = Share Your Gifts

In this article, we’ll explore appreciation, which is a pillar of happiness and one of the fastest ways to shift a student's mood and perspective. The definition of appreciation is "gratitude; thankful recognition." Developing gratitude helps students to focus on what is working in their lives, and also to train their minds to notice the good things that are all around. Learning to appreciate even the little things in life, such as a sunny day, a smile or a good meal, improves one's outlook substantially, and helps to develop a more optimistic and resilient attitude. What we focus on is what grows -- and gratitude promotes positivity.

Continue reading here

How evil a thing is inflation?

noreply@blogger.com (Abdulla Nafiz) — Fri, 05 Oct 2012 17:58:00 +0000

It is true that in times of general rise in the price level, if all groups of prices, such as agricultural prices, industrial prices, prices of minerals, wages, rent and profit rise in the same direction and by the same extent, there will be no net effect on any section of people in the community. For example, if the prices of goods and services, which a worker buys rises by 50 per cent and if the wage of the worker also rises by 50 per cent then there is no change in the real income of the worker, ie., his standard of living will remain constant. However, in practice, all prices do not move in same direction and by same percentage. Hence, some classes of people in the community are affected by inflation more favorably than others. This is explained as follows:

Continue to read full article here

How secure is your DSL modem/router?

noreply@blogger.com (Abdulla Nafiz) — Mon, 01 Oct 2012 19:45:00 +0000

So, you think you're doing a pretty good job in terms of computer security on your home PC? You've kept your computer fully patched against the latest vulnerabilities? You've ensured that your PC is running the latest-and-greatest anti-virus updates?

Good for you.

Now, how about your router?

My suspicion is that the typical computer user doesn't give a second thought about whether their router could be harbouring a security threat, imagining that the devices don't need to be treated with suspicion.

But if you think that, you're quite wrong. Read full article here

Mitigating Risk - Why Sticking with Windows XP is a Bad Idea - IDC

noreply@blogger.com (Abdulla Nafiz) — Sat, 26 May 2012 15:39:00 +0000

This IDC white paper discusses why remaining with Windows XP is no longer a good business decision.

This IDC white paper discusses the benefits associated with a move from Windows XP to Windows 7 and presents a case for the many reasons why remaining with Windows XP is no longer a good business decision. Costs tend to soar when older products are used beyond their intended lifecycle, and IDC research has clearly confirmed that trend.

Download it form here: https://www.microsoft.com/en-us/download/confirmation.aspx?id=29883

Petition to stop inciting hate in Maldives.

noreply@blogger.com (Abdulla Nafiz) — Wed, 14 Mar 2012 03:12:00 +0000

We are calling upon Mr.Nasheed, the leader of Maldivian Democratic Party (MDP) to stop actions which has caused unprecedented divide and hatred among the people of Maldives. After resigning on live television on the 7th of February 2012, ex-president Nasheed, after 24 hours of his resignation then claimed he was forced to resign under duress, yet has failed to prove or take the matters to the courts. Instead he has gathered his supporters to the streets, incited hatred among people and has continued to organize street protests and encourages acts of terror. We urge Mr.Nasheed to stop calling out for civil disobedience and stop encouraging the break of law.

We condemn the acts of violence and arson, which are being committed by MDP and its supporters. We are urging all political parties in the Maldives to find a peaceful resolution to the current situation through civil decorum, without the call for violence and within democratic means. Stop further riots and civil disobedience acts by MDP. We are calling upon all Leaders to intervene in MDP’s terrorism and urge to stop their actions….

To read full details and sign the petition kindly follow this link: http://www.avaaz.org/en/petition/Stop_the_inciting_hate_in_Maldives/?cvGgUcb

Maldives president quits

noreply@blogger.com (Abdulla Nafiz) — Tue, 07 Feb 2012 09:40:00 +0000

The president of the Maldives has resigned, after clashes in the capital, Male, between soldiers and police who sided with anti-government protesters.

Mohamed Nasheed confirmed his resignation on Tuesday in a nationwide broadcast after police seized control of the state television station.

"It will be better for the country in the current situation if I resign. I don't want to run the country with an iron fist. I am resigning," Nasheed said.

A statement issued by the president's office on Tuesday called for people to remain calm.

"The government of Maldives together with all state institutions will work to ensure peace and stability in Male," it said. Read more at http://www.aljazeera.com/news/asia/2012/02/20122765334806442.html

I do not believe that we are in a situation of crisis here in the Maldives though…It was a peaceful power shift for the good of the nation and its people.

On SOPA

noreply@blogger.com (Abdulla Nafiz) — Fri, 20 Jan 2012 04:38:00 +0000

Several readers have asked me my opinion of SOPA, the Stop Online Piracy Act. I fear that in this case, the devil is in the details, so I find it hard to reach a strong view. But I have been disturbed by the relatively knee-jerk reaction of the anti-SOPA crowd. This is a hard issue, and when someone makes it sound easy, I feel like they haven't thought it through very thoroughly.

The anti-SOPA crowd argues that this is a matter of basic liberty. But it's not. In a free society, you don't have the freedom to steal your neighbor's property. And that should include intellectual property. Continue to read at Greg Mankiw's Blog..

The Taj Mahal

noreply@blogger.com (Abdulla Nafiz) — Sat, 24 Dec 2011 04:25:00 +0000

Guest article by Prof Dawn Lovett

The Taj Mahal is undoubtedly one of the finest examples of architecture known to man. It is recognized all around the world as an engineering marvel. Construction began in 1631 but took many years to complete. While most people consider the Taj Mahal to be the white domed building and towers, it actually encompasses a large complex of buildings and gardens that stretches over 55 acres. It is located on the bank of the river Yamuna in the city of Agra, which is about 125 miles south of the capital, New Delhi. Here are some interesting facts about this world marvel:

1. It was built as a mausoleum for the favorite wife of Mughal Emperor Shah Jahan. He loved her so much that he commissioned the mausoleum to be built for her tomb.

2. The design for the Taj Mahal and the surrounding land followed the Mughal tradition of the paradise garden, with abundant trees, flowers, and water features. The symmetrical and geometric elements decorating the mausoleum also come from Mughal design traditions. The Taj Mahal complex itself was meant to replicate the house of the departed in paradise.

3. Everything at the Taj Mahal is symmetrical apart from one feature. The original tomb is located at the very center of the (hall), with Jahan’s larger tomb to one side of it. The placing of the tomb was thought to be conceived by Jahan’s son, who may have done so to spite his father (see number 7).

4. It took 22 years, 22,000 workers, and 1,000 elephants to construct the Taj Mahal. If it was built in present times, it would have cost over $100 million to complete.

5. Several features of the Taj Mahal are not what they seem. The four pillars tilt slightly outwards so that they would fall away from the tomb in an earthquake. The writing around the entryway becomes larger near the top so that it appears to be all one size to a viewer standing at the base.

6. The materials used to construct the Taj Mahal came from all over India and other areas of Asia. The walls are made of brick and rubble and are covered with fine white marble. 28 types of precious stones including jade, sapphire, turquoise, and carnelian were inlaid in the white marble for the various decorations.

7. Shah Jahan planned on building another Taj Mahal as a tomb for himself. It would have been made of black marble and located across the river Yamuna from the original Taj. However, his son prevented him from doing so and imprisoned him in his own palace until his death.

8. It is not known exactly who was the chief architect of the Taj Mahal, but most historians believe it was Ustad Ahamd Lahouri, an architect from Shah Jahan’s court. In fact, many designers and craftsman were involved in the design and construction of the Taj Mahal – even Shah Jahan had an active role in the process.

9. No one knows what was planted in the original Taj Mahal gardens. Contemporary accounts note fruit-bearing trees and aromatic herbs, but nothing specific. The vegetation has changed over the years, especially during the time of British rule.

10. Since Muslims forbid any depiction of Allah, the decoration on the Taj Mahal is limited to various geometric designs and calligraphy, which comes from the Qur’an. Most of the texts refer to themes of judgment and are set in black jasper.

Dawn Lovett has been a professor of engineering for over 20 years. She also owns the site Online Engineering Degrees for students interested in getting an online degree in engineering.

Good Teaching and Bad Business Writing

noreply@blogger.com (Abdulla Nafiz) — Mon, 21 Nov 2011 19:51:00 +0000

Does high school teach us to write in a way that’s counterproductive in the real world? Most high schools teach us to write in a formal, literary style, usually by studying the ‘classics.’

But does this make sense in today’s economy? When you go out and find your first job, what’s the first thing you do?

Change your writing style! You develop a new writing style that goes against many of the rules you were taught in high school.

To develop your (and your kid’s) career, wouldn’t it make sense to learn business writing skills instead of elegant, academic writing styles?

Of course, it’s not the teacher’s fault. They’re following the curriculum. But, if you had a magic wand, what would you change?

Here’s some ideas.

Kindly follow this link at klariti.com to read the full article.

Wastage of Human Resources

noreply@blogger.com (Abdulla Nafiz) — Mon, 14 Nov 2011 08:33:00 +0000

The term wastage of human resources is the loss of employees other than by redundancy. Redundancy points out the position of the employee as no longer needed for any available job and therefore liable to dismissal. In an organization, the wastage of human resources is mainly caused by the factors Labor turnover, Idle Time and Absenteeism.

Being Human- Kindness is the Key

noreply@blogger.com (Abdulla Nafiz) — Mon, 31 Oct 2011 19:15:00 +0000

Kindness is a virtue for some and those who acquire it live a contented life there after. But our Business Managers and leaders are forgetting the art of munificence be it, termination of employment (where it can be avoided), not giving any leaves, discourteous language, unfriendly behavior, ego, being abusive, and what not, our forum (citehr) keeps receiving such queries…

What do you think is missing amongst other things? Here is what I feel can bring some relief….

“The ideals which have lighted my way and time after time have given me new courage to face life cheerfully, have been Kindness, Beauty, and Truth. The trite subjects of human efforts, possessions, outward success, and luxury have always seemed to me contemptible” - Albert Einstein

more at http://www.citeman.com/16663-being-human-kindness-is-the-key.html#ixzz1cO8v5SGn

4 Technologies to Change The World

noreply@blogger.com (Abdulla Nafiz) — Mon, 31 Oct 2011 10:03:00 +0000

Right now is such an exciting time to be alive, as science and technology hurtles humanity forward at such incredible rates. Who would’ve thought ten years ago that we’d have one touch access to all the information in the world, on our phones!? At this rate, what are we going to see in another ten years?

With any luck, it’ll be one or more of these four incredible technologies that will very likely change the world.

Continue reading at http://www.makeuseof.com/tag/4-technologies-change-world/

Engineering and Management

noreply@blogger.com (Abdulla Nafiz) — Tue, 25 Oct 2011 21:04:00 +0000

A job oriented job was possible if you attained an engineering qualification some 4 decades ago. Engineering is in various types of fields such as Civil, Mechanical, Electrical, Electronics, Mining, Aeronautical, Environment and perhaps few other fields. Engineering course is a four year course including computer subjects.

Only an engineering qualification now is not enough and you have to additionally do an MBA to build a career. The Engineering and manufacturing sector will take a worldwide stage and will be very prominent in the world. Experts say with management qualification career choices will open up for job aspirants and that they will have a range of choices to excel.

This situation persists, thanks to high job satisfaction ratings due to the profession’s financial security and highly creative, intellectual environment; not to forget the rewards of being able to benefit society through scientific and technological discovery.

The engineering industry has always had a requirement for high quality talent. Firstly it did not appear as if it was booming because other sectors were recruiting in larger numbers. Today, the scenario has changed. Therefore, in comparison, it seems as if there is a boom in the engineering industry. Secondly, there was large scale migration of talent from engineering industry to other industries, notably IT. This necessitated recruitment of replacement in fairly substantial numbers.

more at http://www.citeman.com/16806-engineering-and-management.html#ixzz1bpVOKeNp

Quality of Work Life (QWL)

noreply@blogger.com (Abdulla Nafiz) — Fri, 21 Oct 2011 06:54:00 +0000

The term Quality of Work Life (QWL) aims at changing the entire organizational climate by humanizing work, individualizing organizations and changing the structural and managerial systems. It takes into consideration the socio-psychological needs of the employees. It seeks to create such a culture of work commitment in the organizations which will ensure higher productivity and greater job satisfaction for the employees.

Quality of work life refers to the favorableness or unfavorableness of the job environment of an organization for its employees. It is generic term which covers a person’s feelings about every dimension of his work e.g. economic incentives and rewards, job security, working conditions, organizational and interpersonal relationships etc. The term QWL has different meanings for different people. A few important definitions of QWL are as follows:

Kindly continue to read full article at MBA Knowledge Base

What is Snap in Windows 7?

noreply@blogger.com (Abdulla Nafiz) — Sat, 24 Sep 2011 16:32:00 +0000

http://windowsteamblog.com/windows/b/windowsexperience/archive/2011/09/23/what-is-snap-in-windows-7.aspx

The Six Thinking Hats

noreply@blogger.com (Abdulla Nafiz) — Thu, 22 Sep 2011 02:13:00 +0000

Completing the square

noreply@blogger.com (Abdulla Nafiz) — Sun, 04 Sep 2011 06:37:00 +0000

I have been thinking of posting some of my Mathematics and Physics lessons on this blog. These are lessons that, I have been using since 1997 during my private tutoring hours, improving over time, mostly for students studying for London General Certificate of Education, Ordinary Level [London GCE O’ Level] and Cambridge International General Certificate of Secondary Education, Ordinary Level [CIE IGCSE O’ Level]. As the very first Mathematics tutorial on this blog, today I am going to show how quadratic equations can be solved by employing completing the square method.

A few assumptions

Before we begin, a few assumptions need to be made regarding the prerequisites for this tutorial as, it so happens that, without prior knowledge of elementary algebra, it will be a bit difficult to grasp ideas that will be conveyed in this lesson. Thus, I will simply assume that, the reader is familiar with and has basic knowledge and understanding of

Algebra
The concept of roots, more specifically square roots
Expansion of binomials and factoring of trinomials using the two identities:
$(a + b)^{2} \equiv a^{2} + 2 a b + b^{2}$

$(a - b)^{2} \equiv a^{2} - 2 a b + b^{2}$
Quadratic equations including their general form

Completing the square

Before going in to solving equations, let us first see what completing the square is all about. For that, consider a simple trinomial and see how it can be changed to a perfect square by applying our knowledge of the first identity above. Consider this quadratic expression $x^{2} + 4 x + 3$

Now, how can we turn this into a perfect square, just like the identity? By careful observation it can be seen that, the constant or the last term of a perfect square is always the square of one half the coefficient of the x term. So, it must have 4 as the constant term in this expression. But, it has a 3! How do we change the 3 into 4? Simple! By adding 1. But, by adding 1 to the expression, also changes it to a completely new one changing its meaning. To overcome that, we need to subtract one from it.

So that, the expression retains its original meaning and would now look like this: $x^{2} + 4 x + 4 - 1$ Now, ignoring –1, the first three terms of this expression is a perfect square trinomial, which can be expressed as a square of the binomial: $(x + 2)^{2}$

Therefore, the complete expression now looks like this: $(x + 2)^{2} - 1$

This process of turning a quadratic expression into a complete square is, what really completing the square means.

Solution of quadratic equations by completing the square

Let us consider the above example by equating it to zero, making it a quadratic equation.
$x^{2} + 4 x + 3 = 0$
Going through the same steps as above we get
$(x + 2)^{2} - 1 = 0$
Solving for $x$ ,
$x = &PlusMinus; 1 - 2$
Giving $x = - 1$ and $x = - 3$

Note that, one can also get the same results by factoring using the difference of two squares identity after completing the square as, 1 is a square number.

One more example using the second identity

Consider this quadratic equation: $x^{2} - 6 x - 7 = 0$ To complete the square we need to change the constant term to 9. To do that we add 16 to it and subtract 16 to keep the whole expression unchanged giving: $x^{2} - 6 x + 9 - 16 = 0$

Taking -16 to RHS and completing square on LHS $(x - 3)^{2} = 16$ solve for x by taking roots on both sides $x - 3 = &PlusMinus; \sqrt{16}$

Therefore, $x = 7$ and $x = - 1$

Another example

In the two examples above coefficient of the first term is 1. Let us now see an example with a coefficient greater than 1.

Consider

3 x^{2} - 36 x - 39 = 0

This time we will take the constant term to RHS and complete the square on LHS

3 x^{2} - 36 x = 39

In order to make the coefficient of the first term one, divide the whole equation by 3

x^{2} - 12 x = 13

Half the coefficient of middle term is -6 squaring it gives 36. This needs to be added to both sides of the equation.

x^{2} - 12 x + 36 = 13 + 36

Simplifying

x^{2} - 12 x + 36 = 49

Factoring LHS

(x - 6)^{2} = 49

Solving for x gives, x = 13 and x = -1

That is all about it, for now. Good Luck!