SMK AMINUDDIN BAKI JOHOR BAHRU.

Name : MOHAMMAD NORHAMIZAN BIN AHMADClass : 5 CERDASTeacher : Mrs. CHEW SIEW FANG

Icnumber: 951016-01-5981

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CONTENTAcknowledgement……………..………………………….

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Objective……………………………..…………………….

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Introduction…………………………..…………………….

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Part A……………………………………………………….

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Part B…………………………………………………………..

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Part C……………………………………………………….

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Reflection………………………………………………….

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Conclusion………………………………………………………………

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ACKNOWLEDGEMENTFirst of all, I would like to say Alhamdulillah, thanks to God for giving me the work. Their advices, which I really needed to motivate myself in completing this project. They also had been supporting and encouraging me to complete this task soon as I could so that I would not procrastinate in doing so.

Then, I would like to thank my Additional Mathematics teacher, Mrs Chew Siew Fang, who had been the one to guide me and the whole class throughout this project. Even though I had some difficulties in finishing this task, I managed to finish it well as she taught me patiently till I got her point.

I want to thank the media, for providing me countless informations on this topic. I would be lost if there’s no internet at my home.

Thanks to my friends who had been always supporting me. Even though this project had to be done individually, we discussed with each other on anything that was related to this project via Twitter and text messages. We shared ideas and methods to answer those asked questions correctly.

Last but not least, thanks to anyone who had been contributed either directly or indirectly in completing this project work. Without them, I believed, this project work could not be done in such a good way.strength and health to do this project work successfully and finish it on time.

Not forgotten, thanks to my parents for providing everything, such as laptop and internet connection, which really was a big help to me in finishing this project up as I could surf the net to find information and guidance for me to make it

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OBJECTIVEThe aims of carrying out this project work are:

To apply and adapt a variety of problem-solving st

rategies to solve problems.

To improve thinking skills.

To promote effective mathematical communicatio

n.

To develop mathematical knowledge through prob

lem solving in a way that increases students’

interest and confidence.

To use the language of mathematics to express m

athematical ideas precisely.

To provide learning environment that stimulates a

nd enhances effective learning.

To develop positive attitude towards mathematics.

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INTRODUCTIONHistory of Equations.

It is often claimed that the Babylonians (about 400 BC) were the first to solve quadratic equations. This is an over simplification, for the Babylonians had no notion of 'equation'. What they did develop was an algorithmic approach to solving problems which, in our terminology, would give rise to a quadratic equation. The method is essentially one of completing the square. However all Babylonian problems had answers which were positive (more accurately unsigned) quantities since the usual answer was a length.

The Arabs did not know about the advances of the Hindus so they had neither negative quantities nor abbreviations for their unknowns. However al-Khwarizmi (c 800) gave a classification of different types of quadratics (although only numerical examples of each). The different types arise since al-Khwarizmi had no zero or negatives. He has six chapters each devoted to a different type of equation, the equations being made up of three types of quantities namely: roots, squares of roots and numbers i.e. x, x2 and numbers.

1. Squares equal to roots.2. Squares equal to numbers.3. Roots equal to numbers.4. Squares and roots equal to numbers, e.g. x2 + 10x = 39.5. Squares and numbers equal to roots, e.g. x2 + 21 = 10x.6. Roots and numbers equal to squares, e.g. 3x + 4 = x2.

Al-Khwarizmi gives the rule for solving each type of equation, essentially the familiar quadratic formula given for a numerical example in each case, and then a proof for each example which is a geometricalcompleting the square.

Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book Liber embadorum published in 1145 which is the first book published in Europe to give the complete solution of the quadratic equation.

A new phase of mathematics began in Italy around 1500. In 1494 the first edition of Summa de arithmetica, geometrica, proportioni et proportionalita, now known as the Suma, appeared. It was written by Luca Pacioli although it is quite hard to find the author's name on the book, Fra Luca appearing in small print but not on the title page. In many ways the book is more a summary of knowledge at the time and makes no major advances.

Pacioli does not discuss cubic equations but does discuss quartics. He says that, in our notation, x4 = a + bx2 can be solved by quadratic methods but x4 + ax2 = b and x4 + a = bx2

are impossible at the present state of science.

Scipione dal Ferro (1465-1526) held the Chair of Arithmetic and Geometry at the University of Bologna and certainly must have met Pacioli who lectured at Bologna in 1501-2. Dal Ferro is credited with solving cubic equations algebraically but the picture is somewhat more complicated. The problem was to find the roots by adding, subtracting, multiplying, dividing

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and taking roots of expressions in the coefficients. We believe that dal Ferro could only solve cubic equation of the form x3 + mx = n. In fact this is all that is required.

For, given the general cubic y3 - by2 + cy - d = 0, put y = x + b/3 to get x3 + mx = n where m = c - b2/3, n = d - bc/3 + 2b3/27.

However, without the Hindu's knowledge of negative numbers, dal Ferro would not have been able to use his solution of the one case to solve all cubic equations. Remarkably, dal Ferro solved this cubic equation around 1515 but kept his work a complete secret until just before his death, in 1526, when he revealed his method to his student Antonio Fior.

Application of Equations.

We now fastforward 1000 years to the Ancient Greeks and see what they made of quadratic equations. The Greeks were superb mathematicians and discovered much of the mathematics we still use today. One of the equations they were interested in solving was the (simple) quadratic equation:

     

They knew that this equation had a solution. In fact it is the length of the hypotenuse of a right angled triangle which had sides of length one.

It follows from Pythagoras’ theorem that if a right-angled triangle has shorter sides   and   and hypotenuse  then

     

Putting   and   then  . Thus 

So, what is   in this case? Or, to ask the question that the Greeks asked, what sort of number is it? The reason that this mattered lay in the Greek’s sense of proportion. They believed that all numbers were in proportion with each other. To be precise, this meant that

all numbers were fractions of the form   where   and   are whole numbers. Numbers like

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1/2, 3/4 and 355/113 are all examples of fractions. It was natural to expect that  was also a fraction. The huge surprise was that it isn’t. In fact

     

where the dots   mean that the decimal expansion of   continues to infinity without any discernible pattern. (We will meet this situation again later when we learn about chaos.)

 was the first irrational number (that is, a number which is not a fraction, or rational), to

be recognised as such. Other examples include  , ,   and in fact "most" numbers. It took until the 19th century before we had a good way of thinking about these numbers. The

discovery that   was not a rational number caused both great excitement (100 oxen were sacrificed as a result) and great shock, with the discoverer having to commit suicide. (Let this be an awful warning to the mathematically keen!) At this point the Greeks gave up algebra and turned to geometry.

Far from being an obscure number, we meet   regularly: whenever we use a piece of A4 paper. In Europe, paper sizes are measured in A sizes, with A0 being the largest with an area of  . The A sizes have a special relationship between them. If we now do a bit of origami, taking a sheet of A1 paper and then folding it in half (along its longest side), we get A2 paper. Folding it in half again gives A3, and again gives A4 etc. However, the paper is designed so that the proportions of each of the A sizes is the same - that is, each piece of paper has the same shape.

We can pose the question of what proportion this is. Start with a piece of paper with sides xand y with x the longest side. Now divide this in two to give another piece of paper with sidesy and x/2 with now y being the longest side. This is illustrated to the right.

The proportions of the first piece of paper are   and those of the second are   

or  . We want these two proportions to be equal. This means that

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or

     

Another quadratic equation! Fortunately it's one we have already met. Solving it we find that

     

This result is easy for you to check. Just take a sheet of A4 (or A3 or A5) paper and measure the sides. We can also work out the size of each sheet. The area   of a piece of A0 paper is given by

     

But we know that   so we have another quadratic equation for the longest side   of A0, given by

     

This means that the longest side of A  is given by   (why?) and that of A  

by  . Check these on your own sheets of paper.Paper used in the United States, called foolscap, has a different proportion. To see why, we return to the Greeks and another quadratic equation. Having caused such grief, the quadratic equation redeems itself in the search for the perfect proportions: a search that continues today in the design of film sets, and can be seen in many aspects of nature.

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After spending countless hours, days and night to finish this project and also sacrificing my time for video games and stuffs during this mid-year school break, there are several things that I can say. I’m going to express it through words anyways.

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After doing some researches, answering the given questions, drawing the graphs and some problem solving, I saw that the usage of quadratic

equation is important in daily life. It is not just widely used in architecture such as determining the area of a sculpture with curve(s) but we use

quadratic equation in our daily life as well. To be related, determining the area is important as it can give the exact amount of the needed cost.

But, what is the use of quadratic equation in daily life of normal people like us? In reality most people are not going to use the quadratic equation in

daily life. Having a firm understanding of the quadratic equation as with most maths helps increasing logical thinking, critical thinking, and number sense.

We use quadratic equations to determine how to shape the mirror of, say a car headlight, that is familiar, and where to put the light. If the light is at the

focus, as it should be, all light from the bulb will be reflected straight out.

As a conclusion, quadratic equation is a daily life essentiality. If there is no quadratic equation, architect won’t be able to create such perfect buildings,

and light from bulbs in front of a car cannot shine brilliantly.

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