3+2√7 512-65 -=2,123, exact to within 0,001. REMARK. Expressions of this kind might be calculated by ap. proximating 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 form a precise idea of 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 the approximation is made. The principles for the extraction of the square root of particular numbers and of algebraic quantities, being established, we will proceed to the resolution of problems of the second degree. Examples in the Calculus of Radicals. 1. Reduce √125 to its most simple terms. 4. Reduce √(x2-ax2) to its most simple terms. 5. Required the sum of √72 and 128 8. Required the sum of 2 Va2b and 3 √64bx 11. Required the product of 5 18 and 3√5. 45 Of Equations of the Second Degree. 137. When the enunciation of a problem leads to an equation of the form ax2=b, in which the unknown quantity is multiplied by itself, the equation is said to be of the second degree, and the principles established in the two preceding chapters are not sufficient for the resolution of it; but since by dividing the two members by a, it b becomes 2=, we see that the question is reduced to finding the square root of a b a 138. Equations of the second degree are of two kinds, viz. equa. tions involving two terms, or incomplete equations, and equations involving three terms, or complete equations. The first are those which contain only terms involving the square of the unknown quantity, and known terms; such are the equations, These are called equations involving two terms because they may be reduced to the form ax=b, by means of the two general transformations (Art. 90 & 91). For, let us consider the second equation, which is the most complicated; by clearing the fractions it be comes 8x2-72+10x2-7-24x2+299, or transposing and reducing 42x2-378. Equations involving three terms, or complete equations, are those which contain the square, and also the first power of the unknown quantity, together with a known term; such are the equations They can always be reduced to the form ax2+bx=c, by the two transformations already cited. Of Equations involving two terms. 139. There is no difficulty in the resolution of the equation b a' ax=b. We deduce from it r2=-, whence x= b b a When is a particular number, either entire or fractional, a we can obtain the square root of it exactly, or by approximation. b If is algebraic, we apply the rules established for algebraic a quantities. But as the square of +m or -m, is +m2, it follows that either of these values in the equation ax2=b, it becomes and b ax(-a)=b, or ax=0. For another example take the equation 4x2-7=3x2+9; by transposing, it becomes, 2=16, whence x=± √16=±4. We have already seen (Art. 138.), that this equation reduces to 378 42 42x2-378, and dividing by 42, x2= -=9; hence x=±3. Lastly, from the equation 3x=5; we find x=±√=√15. 3 3 As 15 is not a perfect square, the values of a can only be deter. mined by approximation. Of complete Equations of the Second Degree. 140. In order to resolve the general equation ax2+bx-c. we begin by dividing both members by the co-efficient of 2, which gives, Now, if we could make the first member (x2+px) the square of a binomial, the equation might be reduced to one of the first degree, by simply extracting the square root. By comparing this member with the square of the binomial (x+a), that is, with x2+2ax+a3, it is plain that x2+px is composed of the square of a first term x, plus the double product of this first term æ by a second, which must 2 be P P 2' , since px=2x; -x; therefore, if the square added to x2+px, the first member of the equation will become the square of x+ -; but in order that the equality may not be destroy p 2 2 ed p must be added to the second member. 4 By this transformation, the equation x2+px=q becomes The double sign ± is placed here, because either ++, or -, squared gives + q 4 q+ From this we derive, for the resolution of complete equations of the second degree, the following general RULE. After reducing the equation to the form x2+px=q, add the square of half of the co-efficient of x, or of the second term, to both mетbers; then extract the square root of both members, giving the double sign ± to the second member; then find the value of x from the re. sulting equation. This formula for the value of x may be thus enunciated. The value of the unknown quantity is equal to half the co-efficient |