Which of the following can be used to find the quadratic equation given the sum and product of roots?

𝑥 2 - (sum of roots)𝑥 + (product of roots) = 0

𝑥 2 + (sum of roots)𝑥 - (product of roots) = 0

𝑥 2 - (sum of roots)𝑥 - (product of roots) = 0

𝑥 2 + (sum of roots)𝑥 + (product of roots) = 0

Nature of Roots/ Sum and Product of Roots Of QE

Nature of the roots and sum& product of the roots of quadratic equations

Quiz on Discriminant and Nature of Roots

SUMMATIVE TEST IN MATHEMATICS 9 (Nature of Roots and Problems Involving Quadratic Equations)

Quadratic Equation 4. Solve quadratic equation using diff methods 5. Determine the nature of roots of equation

Quadratic Equation (Nature of the Roots)

All living things are made up of one or more cells. A cell is the smallest unit that can carry on all of the processes of life. Beginning in the 17th century, curious naturalists were able to use microscopes to study objects too small to be seen with the unaided eye. Their studies led them to propose the cellular basis of life. Hooke In 1665, English scientist Robert Hooke studied nature by using an early light microscope, such as the one in Figure 4-1a. A light micro- scope is an instrument that uses optical lenses to magnify objects by bending light rays. Hooke looked at a thin slice of cork from the bark of a cork oak tree. “I could exceedingly plainly perceive it to be all perforated and porous,” Hooke wrote. He described “a great many little boxes” that reminded him of the cubicles or “cells” where monks live. When Hooke focused his microscope on the cells of tree stems, roots, and ferns, he found that each had similar little boxes. The drawings that Hooke made of the cells he saw are shown in Figure 4-1b. The “little boxes” that Hooke observed were the remains of dead plant cells, such as the cork cells shown in Figure 4-1c. SECTION 1 OBJECTIVES ● Name the scientists who first observed living and nonliving cells. ● Summarize the research that led to the development of the cell theory. ● State the three principles of the cell theory. ● Explain why the cell is considered to be the basic unit of life. VOCABULARY cell cell theory Robert Hooke used an early microscope (a) to see cells in thin slices of cork. His drawings of what he saw (b) indicate that he had clearly observed the remains of cork cells (300) (c). FIGURE 4-1 (a) (b) (c) Copyright © by Holt, Rinehart and Winston. All rights reserved. 70 CHAPTER 4 Leeuwenhoek The first person to observe living cells was a Dutch trader named Anton van Leeuwenhoek. Leeuwenhoek made microscopes that were simple and tiny, but he ground lenses so precisely that the magnification was 10 times that of Hooke’s instruments. In 1673, Leeuwenhoek, shown in Figure 4-2a, was able to observe a previ- ously unseen world of microorganisms. He observed cells with green stripes from an alga of the genus Spirogyra, as shown in Figure 4-2b, and bell-shaped cells on stalks of a protist of the genus Vorticella, as shown in Figure 4-2c. Leeuwenhoek called these organisms animalcules. We now call them protists. THE CELL THEORY Although Hooke and Leeuwenhoek were the first to report observ- ing cells, the importance of this observation was not realized until about 150 years later. At this time, biologists began to organize information about cells into a unified understanding. In 1838, the German botanist Matthias Schleiden concluded that all plants were composed of cells. The next year, the German zoologist Theodor Schwann concluded the same thing for animals. And finally, in his study of human diseases, the German physician Rudolf Virchow (1821–1902) noted that all cells come from other cells. These three observations were combined to form a basic theory about the cel- lular nature of life. The cell theory has three essential parts, which are summarized in Table 4-1. Anton van Leeuwenhoek (1632–1723) is shown here with one of his hand-held lenses (a). Leeuwenhoek observed an alga of the genus Spirogyra (b) and a protist of the genus Vorticella (c). FIGURE 4-2 TABLE 4-1 The Cell Theory All living organisms are composed of one or more cells. Cells are the basic units of structure and function in an organism. Cells come only from the reproduction of existing cells. (a) (b) (c) www.scilinks.org Topic: Cell Theory Keyword: HM60241 mb06se_csfs01.qxd 5/18/07 10:54 AM Page 70

Southeast Asia, vast region of Asia situated east of the Indian subcontinent and south of China. It consists of two dissimilar portions: a continental projection (commonly called mainland Southeast Asia) and a string of archipelagoes to the south and east of the mainland (insular Southeast Asia). Extending some 700 miles (1,100 km) southward from the mainland into insular Southeast Asia is the Malay Peninsula; this peninsula structurally is part of the mainland, but it also shares many ecological and cultural affinities with the surrounding islands and thus functions as a bridge between the two regions. Mainland Southeast Asia is divided into the countries of Cambodia, Laos, Myanmar (Burma), Thailand, Vietnam, and the small city-state of Singapore at the southern tip of the Malay Peninsula; Cambodia, Laos, and Vietnam, which occupy the eastern portion of the mainland, often are collectively called the Indochinese Peninsula. Malaysia is both mainland and insular, with a western portion on the Malay Peninsula and an eastern part on the island of Borneo. Except for the small sultanate of Brunei (also on Borneo), the remainder of insular Southeast Asia consists of the archipelagic nations of Indonesia and the Philippines. Southeast Asia stretches some 4,000 miles at its greatest extent (roughly from northwest to southeast) and encompasses some 5,000,000 square miles (13,000,000 square km) of land and sea, of which about 1,736,000 square miles is land. Mount Hkakabo in northern Myanmar on the border with China, at 19,295 feet (5,881 meters), is the highest peak of mainland Southeast Asia. Although the modern nations of the region are sometimes thought of as being small, they are—with the exceptions of Singapore and Brunei—comparatively large. Indonesia, for example, is more than 3,000 miles from west to east (exceeding the west-east extent of the continental United States) and more than 1,000 miles from north to south; the area of Laos is only slightly smaller than that of the United Kingdom; and Myanmar is considerably larger than France. All of Southeast Asia falls within the tropical and subtropical climatic zones, and much of it receives considerable annual precipitation. It is subject to an extensive and regular monsoonal weather system (i.e., one in which the prevailing winds reverse direction every six months) that produces marked wet and dry periods in most of the region. Southeast Asia’s landscape is characterized by three intermingled physical elements: mountain ranges, plains and plateaus, and water in the form of both shallow seas and extensive drainage systems. Of these, the rivers probably have been of the greatest historical and cultural significance, for waterways have decisively shaped forms of settlement and agriculture, determined fundamental political and economic patterns, and helped define the nature of Southeast Asians’ worldview and distinctive cultural syncretism. It also has been of great importance that Southeast Asia, which is the most easily accessible tropical region in the world, lies strategically astride the sea passage between East Asia and the Middle Eastern–Mediterranean world. Within this broad outline, Southeast Asia is perhaps the most diverse region on Earth. The number of large and small ecological niches is more than matched by a staggering variety of economic, social, and cultural niches Southeast Asians have developed for themselves; hundreds of ethnic groups and languages have been identified. Under these circumstances, it often is difficult to keep in mind the region’s underlying unity, and it is understandable that Southeast Asia should so often be treated as a miscellaneous collection of cultures that simply do not quite fit anywhere else. Roofs of the Forbidden City, Beijing, China Britannica Quiz All About Asia Yet from ancient times Southeast Asia has been considered by its neighbors to be a region in its own right and not merely an extension of their own lands. The Chinese called it Nanyang and the Japanese Nan’yō, both names meaning “South Seas,” and South Asians used such terms as Suvarnabhūmi (Sanskrit: “Land of Gold”) to describe the area. Modern scholarship increasingly has yielded evidence of broad commonalities uniting the peoples of the region across time. Studies in historical linguistics, for example, have suggested that the vast majority of Southeast Asian languages—even many of those previously considered to have separate origins—either sprang from common roots or have been long and inseparably intertwined. Despite inevitable variation among societies, common views of gender, family structure, and social hierarchy and mobility may be discerned throughout mainland and insular Southeast Asia, and a broadly common commercial and cultural inheritance has continued to affect the entire region for several millennia. These and other commonalities have yet to produce a conscious or precise Southeast Asian identity, but they have given substance to the idea of Southeast Asia as a definable world region and have provided a framework for the comparative study of its components.

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