283. The form a+bV-1 is the typical form of imaginary expressions. Reduce to the typical form 6+ √-8. This may be written 6+ √8 × √−1, or 6+2√2 × √−1; here a = 6, and b = 2√2. 284. Two expressions of the form a+b√—1, a−b√1, are called conjugate imaginaries. To find the sum and product of two conjugate imaginaries: From the above it appears that the sum and product of two conjugate imaginaries are both real. 285. THEOREM 1. An imaginary expression cannot be equal to a real number. Since 2 and (c- a)2 are both positive, we have a negative number equal to a positive number, which is impossible. 286. THEOREM 2. If two imaginary expressions are equal, the real parts are equal and the imaginary parts are equal. For, let Then, Square, a+b√=1=c+d√-1. (bd)√1=ca; − (b − d)2 = (c − a)2, which is impossible unless b = d and a = c. 287. THEOREM 3. If x and y are real and x+y√−1=0, then x = 0 and y = 0. which is true only when x = 0 and y = 0. Operations with Imaginaries. 1. Add 5+7V-1 and 8-9√-1. The sum is 5+8+7√-1-9√-1, or 13-2√ 1. 2. Multiply 3+2 √−1 by 5 - 4 √— 1. 3. Divide 14+5V-1 by 2-3√-1. 14 + 5√= 1 _ (14 + 5√ − 1) (2 + 3√ − 1) = v 39. √25 by — 64. 41. 3√3 by 2√−2. 40. √(a+b) by √(a−b). 42. – 5√-2 by 2√− 5. 43. √−2+ √−3 by √− 4 – √− 5. 45. a√ a+b√b by a √-a-b√6. 46. 2√−2+3√− 3 by 3 √—4— 2 √ — 5. CHAPTER XIX. QUADRATIC EQUATIONS. 288. We have already considered equations of the first degree in one or more unknowns. We pass now to the treatment of equations containing one or more unknowns to a degree not exceeding the second. An equation which contains the square of the unknown, but no higher power, is called a quadratic equation. 289. A quadratic equation which involves but one unknown number can contain only: 1. Terms involving the square of the unknown number. 2. Terms involving the first power of the unknown number. 3. Terms which do not involve the unknown number. Collecting similar terms, every quadratic equation can be made to assume the form where a, b, and c are known numbers, and x the unknown number. If a, b, c are numbers expressed by figures, the equation is a numerical quadratic. If a, b, c are numbers represented wholly or in part by letters, the equation is a literal quadratic. In the equation ax2 + bx + c = 0, a, b, and c are called the coefficients of the equation. The third term c is called the constant term. |