108. Let a and c 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. 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, 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 = √245 = 15 to within less than 1. Therefore, 21 15 to within less than 1 4 = 9 to within less than 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, and divide the result by 4. Take the expression, 7√5 √11+√3 and find its value to within less than 0.01. Hence, we have 3.10 for the required result. This is true t 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. 2. Reduce We observe that 25 will divide the numerator, and hence, 50 to its simplest form. 147 Divide the coefficient of the radical by 3, and mu tiply the num ber under the radical by the square of 3; then, CHAPTER VI. EQUATIONS OF THE SECOND DEGREE. 110. Equations of the second degree may involve but one 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. factoring, b2x abd bd2; (ad-bcd)x2-(bcd + b2)x = abd — bd2; dividing both members by the co-efficient of x2, If we now replace the co-efficient of second member by q, we shall have x2 + 2px = 9; x by 2p, and the 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 the form by the following x2 + 2px = 9, |