In (3) the sign following the equality sign is what is known as the double sign, meaning plus or minus. It must be placed before every square root obtained in the solution of a quadratic. 356. Problems. Determine the numerical value of x in the following equations: 357. How a Complete Quadratic is Solved. There are several ways of solving a complete quadratic, only one of which will be shown here. This method is called solution by completing the square. What this method is and why it is so called is illustrated and explained below: (a) 3x2+2x=8 (1) x2+ 2x=8 Div. (a) by 3 (2) x2+3x+(})2=+=35 Add to (1) (3) x+II = ±? Sq. Root (2) Observe that the given equation was arranged in the order of the descending powers of x, with unknowns in the first member and knowns in the second member; the coefficient of x2 was made unity by dividing equation (a) by 3; the first member of (2) was made a square by adding coefficient of x2 2 to both members; the square root of the equation was extracted, giving (3), which was solved as a simple equation. Whenever a complete quadratic is reduced to the form of equation (1) in which the coefficient of x2 is unity, the first member may be made a perfect square by the addition of (coef. of x)2 as in (2). It is then possible to transform the quadratic into a simple equation by extraction of the square root as in (3). 358. Rules for Solution of a Complete Quadratic. (1) Collect the unknowns into the first member in the order of the descending powers; collect knowns into the second member. (2) Divide the entire equation by the coefficient of the second power of the unknown with its sign. (3) Add to both members (coef. of x)2, indicating the addition in the first member and performing the addition in the second member. (4) Extract the square root of both members, placing the double sign before the root of the second member; solve the resulting simple equation, using first the plus sign and then the minus sign. 359. Examples. Apply the rules of the preceding paragraph to the solution of the following equations. Do not try to commit the rules to memory, but refer to them in the book. Be sure to complete the operations required by each rule before applying the next rule. CHAPTER XIV LOGARITHMS SECTION 1. INTRODUCTION. SECTION 2. USE OF THE TABLE. SECTION 3. COMPUTATION BY LOGARITHMS. § 1. INTRODUCTION 360. Illustrations of a Logarithm. If a series of integral powers of 10 be written beginning with 104 and ending with 10-4 we have 104=10000 102 = 100 100 = 1 10 1.1 10-2=.01 10-3=.001 10-4 = .0001 Observe that as we go down the column the exponents of 10 decrease by 1 and that each number obtained from the successive powers of 10 is therefore the preceding number. All these exponents of 10 are logarithms. 10000 may be obtained from 10 by raising 10 to the 4th power. 1000 may be obtained by raising 10 to the 3d power, and so on. We may, therefore, multiply 10000 by 1000 by adding 4 and 3, the exponents of 10, which gives the 7th power of 10. In like manner if we could determine exactly what power of 10, 3250 is, and exactly what power of 10, 8628 is, we could determine the product 3250X8628 by adding those powers of 10 and by writing down the number which the resulting power of 10 equals. By reference to the powers of 10 at the beginning of this paragraph, it will be seen that 3250 and 8628 are greater than the 3d power of 10 and less than the 4th power. Therefore What is wanted, therefore, is a means of determining the decimal which must be added to the 3d power in each instance. These decimal parts of the powers of 10 have been determined and tabulated for all numbers used in computation as explained in the next paragraph. Such a tabulation is called a table or system of logarithms. The number from whose powers other numbers are obtained is called the base of the system. 361. Definition of a Logarithm. A logarithm is the exponent of the power to which the base of the system must be raised to give any other number. |