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=PHASE 1 = =[] Brahmagupta= =[] Yang Hui =


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== ==Yang Hui's works are mentioned in the // Wenyan ge shumu // (1441; “Catalogue of the Books of the Ming Imperial Library”) but were long thought to be irreparably lost. Ruan Yuan, compiler of // Chou ren zhuan // (1799; “Biographies of Mathematicians and Astronomers”), first found fragments of Yang’s Xiangjia Jiuzhang Suanfa (1261; “A Detailed Analysis of the Nine Chapters on the Mathematical Procedures”) in a handwritten copy of an imperial Ming Dynasty encyclopedia, and he later discovered in Suzhou a Song Dynasty edition of // Yang Hui suanfa // (1275; “Yang Hui’s Mathematical Methods”). The latter contains three treatises, // Chengchu tongbian benmo // (1274; “Fundament and Periphery for Continuity and Change in Multiplication and Division”), // Tianmu bilei chengchu jiefa // (1275; “Quick Methods for Multiplication and Division in Surveying and Analogous Categories”), and // Xu gu zhai qi suan fa // (1275; “Selection of Strange Mathematical Methods in Continuation of Antiquity”). A collected edition (1378) of these works was transmitted farther to the east, where it was particularly influential. In Korea it was reprinted during the reign of Sejong in 1433, and it was copied again by the Japanese mathematician Seki Takakazu (// c. // 1640–1708). Of another work, // Riyong suanfa // (1262; “Mathematical Methods for Daily Use”), only the preface and a few problems are known.==

==Yang’s // Jiu **zh ** // ang suan **// fa zuan lei // (// c. // 1275; “Reclassification of the Mathematical Procedures in the Nine Chapters”)— compilation and reclassification, with further explanations, of the problems from the Han Dynasty classic and its commentaries, **// Jiuzhang suanshu // (// c. // 100 bc – ad 50; // Nine Chapters on the Mathematical Procedures //)—contains the oldest representation of what is known in the West as Blaise Pascal’s triangle (// see // the figure; also see Binomial Theorem) In the preface Yang say's that he copied an older version, Huangdi Jiuzhang Sunafa. == 

[[image:Yanghui_triangle.gif width="205" height="323"]]
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Brahmagupta was born in 598 CE in Bhinmal city in the state of Rajasthan of north-west India. He probably lived in Bhillamala for most of his life. brahmagupta gave the solution of the general Linear Equation in chapter eighteen of Brahmasphutasiddhanta, He was most influenced by his father Jisnugupta, who was also a mathematician and astronomer. Brahmagupta was a mathematician as well as an astronomer. He is best known for his contributions to Algebra and basic Arithmetic. Some of his works on the Quadratic Equation are shown below. He also contributed to geometry, and trigonometry. He wrote 25 volumes that have to deal with Math and Astronomy. === =18.43 The difference between rupas, when inverted and divided by the difference of the unknowns, is the unknown in the equation. The rupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted. Which is a solution equivalent to, where rupas represents constants. He further gave two equivalent solutions to the general Quadratic Equation, 18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number]. 18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown. Which are, respectively, solutions equivalent to,  =

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PHASE 2
Suppose a farmer has 3700 yards of fencing to enclose a rectangular field. What is the largest area that the farmer can enclose? . A1: 925 yards

== == Q2: The owner of a ranch decides to enclose a rectangular region with 400 feet of fencing. To help the fencing cover more land, he plans to use one side of his barn as part of the enclosed region. What is the maximum area the rancher can enclose? ==

A2:He can enclose 20,00 yards [[image:Photo_157.jpg]]
==Q3: A farmer wishes to enclose a rectangular region bordering a river using 330 feet of fencing. He wants to divide the region into two equal parts using some of the fence material. What is the maximum area that can be enclosed with the fencing? ==

A3: The Maximum are he can enclose is 27,225yds
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PART 2 OF PHASE 2==

Q1: Some fireworks are fired vertically into the air from the ground at an initial velocity of 92 feet per second. Find the highest point reached by the projectile just as it explodes.
=A1: The highest point the rocket reached before exploding was 125ft.= == ==

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===Q2: A ball is thrown vertically upward with an initial velocity of 93 feet per second. If the ball started from a height of 10 feet off the ground, determine the time it will take for the ball to hit the ground.===

[[image:Photo_160.jpg]]
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== =PHASE 3 Q1) Solve by graphing, x^2+4x-45 = 0= = Q2) Solve by factoring= = =

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Q3) [[image:Photo_159.jpg]]
x =5, x= 9 :)


Equation: 6x^2+4x-5=0

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =