a When the fraction is of the form if we multiply both terms by b, the denominator will become rational. Thus, Since the sum of two quantities, multiplied by their difference, is equal to the difference of their squares; if the fraction is of the form a - b+ ve and we multiply both terms by b-vc, the denominator will be made rational, since it will be b2-c. Thus, a b-√e_ab-ave For the same reason, if the denominator is b-c, the multiplier will be b+√c. If the denominator is √b+√c, the multiplier will be —√c; and, if the denominator is √b-√c, the multiplier will be b+vc. C. These different forms may be embraced in the following GENERAL RULE. If the denominator is a monomial, multiply both terms by the radical quantity; but, if it is a binomial, multiply both terms by the given binomial with the sign of one of its terms changed, and the denominator will be rational. Reduce the following fractions to equivalent fractions, having rational denominators. REMARK.--The utility of these transformations, consists in diminishing the amount of calculation, necessary to obtain the numerical value of a fractional radical to any required degree of accuracy. 1 Thus, suppose it is required to obtain the numerical value of the fraction true to six places of decimals. If we make the calculation without rendering the denominator rational, it will be found, that we must first extract the square root of 2, to seven REVIEW.-204. When the denominator of a fraction is either a monomial or a binomial, containing radicals of the second degree, how may it be reduced to a fraction having a rational denominator? SIMPLE EQUATIONS CONTAINING RADICALS. 195 places of decimals, and then divide 1 by this result. But if we render the denominator rational, the calculation merely consists in finding the square root of 2, and then dividing by 2. The work by the latter method, requires only about half the labor of that by the former. Besides, the operator feels certain, if he has made no mistake, that the last figure of his result is correct. Whereas, by the other mode, as the divisor is too small, the quotient figures soon become too large. Thus in this example, if we use seven decimals for a divisor, the seventh figure of the quotient is too large; if we only use six places of decimals, the sixth figure will be erroneous. REMARK. It is proper to notice, that the signs and √ applied to a monomial, both have the same meaning. There is a want of uniformity among the best writers, in the manner of making the radical sign before a monomial. SIMPLE EQUATIONS CONTAINING RADICALS OF THE SECOND DEGREE. NOTE TO TEACHERS. This part of the subject of Equations of the First Degree, could not be treated till after Radicals. It may be omitted entirely by the younger class of pupils. ART. 205.—In the solution of questions involving radicals, much will depend on the judgment of the pupil; but the easiest processes can only be learned from practice, as almost every question can be solved in several ways. The following directions will be frequently found useful. 1st. When the equation contains one radical expression, transpose it to one side of the equation, and the rational terms to the other side; then involve both sides to a power corresponding to the radical sign. Thus, if we have the equation √(x−1)—1—2, to find x. Squaring, --1=9, from which x-10.. 2d. When more than one expression is under the radical sign, the operation must be repeated. Thus, a+x=√(a2+x√c2+x2), to find x. 3d. When there are two radical expressions, it is generally better to make one of them stand alone on one side, before squaring. Thus, (x-5)-3=4-√(x-12), to find x. Transposing,(x-5)=7-√(x-12). Squaring, 2-5-49-14√(x-12)+x-12. Squaring, x-12=9, from which x=21. CHAPTER VII. EQUATIONS OF THE SECOND DEGREE. ART. 206.—An Equation of the Second Degree (See Art. 148), is one in which the greatest exponent of the unknown quantity is 2. Thus, x2-9, and 5x2+3x=26, are equations of the second degree. An equation containing two or more unknown quantities, in which the greatest exponent, or the greatest sum of the exponents of the unknown quantities, is 2, is also an equation of the second degree. Thus, xy-6, x2+xy=8, xy+x+y=11, are equations of the second degree. Equations of the Second Degree, are frequently denominated Quadratic Equations. ART. 207.-Equations of the second degree are of two kindsincomplete and complete. An incomplete equation of the second degree, is of the form ax2-b, and contains only the second power of the unknown quantity, and known terms. Thus, x2=9, and 8x2-5x2-12, are incomplete equations of the second degree. An incomplete equation of the second degree, is frequently denominated a pure quadratic equation. A complete equation of the second degree, is of the form ax2+bx=c, and contains both the first and second powers of the unknown quantity, and known terms. Thus, 3x2+4x=20, and ax2-bx2+ dx-ex=f-g, are complete equations of the second degree. A complete equation of the second degree, is frequently denominated an affected quadratic equation. Give ex REVIEW. 206. What is an equation of the second degree? amples. If an equation contains two unknown quantities, when is it of the second degree? Give examples. 207. How many kinds of equations of the second degree are there? What are they? What is the form of an incomplete equation of the second degree? What does it contain? Give an example. What is the form of a complete equation of the second degree? What does it contain? Give an example. What is a pure quadratic equation? What is an affected quadratic equation? ART. 208.-Every equation of the second degree, may be reduced to one of the forms ax2-b, or ax2+bx=c. For, in an incomplete equation, all the terms containing 2 may be collected together, and then, if the coëfficient of x2 contains more than one term, it may be assumed equal to a single quantity, as a, and the sum of the known quantities, to another quantity, b, and then the equation becomes ax2=b, or ax2-b=0. So a complete equation may be similarly reduced; for all the terms containing 22, may be reduced to one term, as ax2; and those containing x, to one, as bx; and the known terms to one, as c; then the equation is ax2+bx=c, or ax2+bx-c−0. Hence, we infer: That every equation of the second degree, may be reduced to an incomplete equation involving two terms, or to a complete equation involving three terms. Frequent illustrations of these principles will occur hereafter. INCOMPLETE EQUATIONS OF THE SECOND DEGREE. ART. 209.—1. Let it be required to find the value of x in the equation x2-16=0. Transposing, Extracting the square root of both members, Verification. x=4, that is, x=+4, or -4. (+4)2—16=16-16=0. or, (-4)2-16=16—16=0. 2. Find the value of x in the equation 5x2+4=49. Transposing, Dividing, 5x245 x2 9 4. Given ax2+b=cx2+d, to find the value of x. ax2-cx2-d-b or, (a—c)x2-d-b d-b a-c Id-b a |