Theory of Quadratic Equations. Equations in the Form of Quadratics ELEMENTARY ALGEBRA I. INTRODUCTION . 1. Symbols Representing Numbers. Algebra, like arithmetic, treats of numbers; but in arithmetic, numbers are usually represented by means of Arabic numerals, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, while in algebra they are represented also by letters, as a, b, c, x, y, z. Numbers represented by letters are called literal numbers. The student has seen how the rules of arithmetic can be abbreviated by the use of letters to represent numbers. When a rule is expressed by means of letters, the result is a formula. The rule, "the number of square units in the area, A, of a rectangle is equal to the product of the number of units in the length, 1, and the number of units in the width, w," may be expressed by means of the formula: A = 1X w. 1. Find the area of a rectangle whose length is 6 inches and whose width is 4 inches. A=1xw w Substitute 6 for 1 and 4 for w in the formula A = 1 × w. Rectangle Thus, A = 1 × w = 6 × 4 = 24, the number of square inches in the area. 2. Find the area of a rectangle whose length is 12 inches and whose width is 9 inches. 3. Explain the use of the following formulas: Area of a triangle, A =¦× a × b, where b is the number of units in the base and Circumference of a circle, A=1x axb Triangle C = 2 × TX R (π = 3.1416). 2. Symbols Representing Operations. The following table shows that the symbols of operation used in algebra are the same, with few additions, as those used in arithmetic : The sign for equality,=, is used as in arithmetic. Notice that while with arithmetical numbers multiplication is indicated by the sign x, as 3 × 2, or 3 × 5 × 7, with literal numbers the sign is usually omitted. Thus, 3 a, which is read three a, means 3 × a, and 3 ab, which is read three ab, means 3 × a × b. 3a +5 means that 3 times a is to be increased by 5, and is read three a plus five. What does 3 a 5 mean? If a = 2, what is the value of 3 a? of 3 a + 5? of 3 a- 5? ORAL EXERCISE 3. 1. What is meant by 3x? by 5 b? 2. What is meant by 3x + 4? by 3x+5y? 3. If x = 2, what is the value of 5 x? 2, and b 4. If a = b÷a? of 3 ab? = 4, what is the value of 4÷a? of 5. What is meant by 5 ab? 6 xyz? 2 mnp+rs? 2a.3b? 6. What is meant by ab÷c? mnp÷r? xy ÷ 8? 7. When x = 1, and y = 2, find the value of 3 xy; of x + y ; 9. Read m÷n = 5 xy. In what other way can the same statement be written? |