EXAMPLES. 1. 3√5ab × 4√20a = 12,/100a2b = 120a√√b. b 108. Let a and cd represent any two radicals of the second degree, and let it be required to find the quotient of the first by the second. This quotient may be indicated thus, Hence, to divide one radical of the second degree by another, we have the following RULE. Divide the co-efficient of the dividend by the co-efficient of the divisor for a new co-efficient; after this, write the radical sign, placing under it the quotient obtained by dividing the quantity under the radical sign in the dividend by that in the divisor. For example, 5ab. 2b √c= And, 5a 26 C 6bc =3a3c. 12ac √√6bc ÷ 4c √2b = 3a 109. The following transformation is of frequent application in finding an approximate value for a radical expression of a particular form. Having given an expression of the form, in which a and p are any numbers whatever, and q not a per fect square, it is the object of the transformation to render the denominator a rational quantity. This object is attained by multiplying both terms of the frac tion by p√, when the denominator is p+, and by P√7, when the denominator is p√7; and recollecting that the sum of two quantities, multiplied by their difference, is equal to the difference of their squares: hence, in which the denominators are rational. As an example to illustrate the utility of this method of ap. proximation, let it be required to find the approximate value of But, 7 √5 = √ 49 × 5 = √24515 to within less 2115 to within less than 1 4 = 9 to within one fourth. ; hence, 9 differs from the true value by less than If we wish a more exact value for this expression, extract the square root of 245 to a certain number of decimal places, add 21 to this root, ana divide the result by 4. and find its value to within less than 0.01. Now, 7/55=√√55 × 49 = √√2695 = 51.91, within less than 0.01, This is true to Hence, we have 3.10 for the required result. By a similar process, it may be found, that, 3+2√7 == 2.123, is exact to within less than 0.001. REMARK.—The value of expressions similar to those above, may be calculated by approximating to the value of each of the radicals which enter the numerator and denominator. But as the value of the denominator would not be exact, we could not determine the degree of approximation which would be obtained, whereas by the method just indicated, the denominator becomes rational, and we always know to what degree of accuracy the approximation is made. We observe that 25 will divide the numerator, and hence, Divide the coefficient of the radical by 3, and mu tiply the num ber under the radical by the square of 3; then, 15. What is the sum of 18a563 and √50a3b3. Ans. (3a2b+ 5ab) Zab. CHAPTER VI. EQUATIONS OF THE SECOND DEGREE. 110. Equations of the second degree may involve but on. unknown quantity, or they may involve more than one. We shall first consider the former class. 111. An equation containing but one unknown quantity is said to be of the second degree, when the highest power of the unknown quantity in any term, is the second. transposing, adx2 -bcdx2 - bcdx b2x abd - bd2; factoring, dividing both members by the co-efficient of x2, and since every equation of the second degree may be reduced, in like manner, we conclude that, every equation of the second degree, involving but one unknown quantity, can be reduced to |