Q II A 



Q U A 



OuADRATA Legio, among the Romans. See Square 

 Legion. 



QUADRATIC Equation, in yllgelra, is an equation 

 of which t.he hi;^heft power of the unknown quantity riles 

 only to the iL-cond degree. If this power enters alone, it is 

 called 2.ftmple quadratic ; and when the fecond power and 

 fimple quantity both occur, it is termed an adfcBed quadratic 

 equation, thus : 



x' + a X = b, is an adfe3ed quadratic, and 



x' zzL b, or ax' = b, ^Jtmple quadratic equation. 



Some authors clafs all fuch equations as contain two dif- 

 ferent powers of the unknown quantity, the one being the 

 double of the other, under the general term quadratics, fuoli 

 as X* + a .v" — b, x"" + ^.r"" = b, &c. each of which is 

 called a quadratic equation, becaufe tlieir folution depends 

 upon precifely the fame principles as the former. 



Every quadratic may be reduced to the form x' -\- a x =: I; 

 in which it is, however, to be underilood, that a and b may 

 be either pofitive or negative ; and the general folution of it 

 is exprefled by the formula 



--T±^'(J + 



If we gire to a and b all the variety of figns they admit 

 of, quadratic equations may be divided into four dillinft 

 clafles, ■viz, 



x" 4- a X z=: — b 



x' — a X ^=. — b 



x' ^ ax — -\- b 



X- — a X = + b ; 



I. 

 2. 



3- 

 4- 



and their feveral roots will be exhibited by the following 

 formulx, viz. 



2. .V = 



+ 



+ 



+ V 



± V 





See Equations. 



From thefe forms it is obvious that every quadratic equa- 

 tion has two roots arifing out of the ambiguous fign +, 

 prefixed to the fecond member of the root. It is alfo ob- 

 vious, that if in the ift and 2d form, b be greater than - 



4' 

 the two roots in both forms are imaginary, or impoflible ; 



but if b be lefs than — , then they are in each form both 



real, being in the lit both negative, and in the 2d both 

 pofitive ; becaufe the quantity exhibited under the radical 

 form is, in both equations, lefs than that without the radical ; 

 and confequently both the fum and difference will have the 



fame fign as - . 



° 2 



In the 3d and 4th forms, the roots are neceffarily both 

 real, but one pofitive and the other negative ; the 3d form 

 having its greateft root negative, and 4th its greateft root 

 jktfitive ; as is obvious by infpe6lion. 



It is, however, not neceffary to confider quadratic equa- 

 tions under thefe four forms, as the folution of them all may 

 be reduced to one general rule, as exhibited in the preceding 

 part of this article, and which may be given in words as 

 follows. Having reduced the proposed equation to any one 



9 



of tiie above forms, the roots of it will be equal to iialf th/? 

 co-efficient of the fecond term, with its fign changed ; plu; 

 and minus the fquare root of the fquare of that half co- 

 efficient prefixed to b, with its proper fign, whether plus or 

 minus. Tiie principles on which the preceding formulae 

 are obtained, are thefe, viz. (x + ~a)' -^ x' + a x -f 3a'; 

 therefore, if an equation be propofed under the form 

 x' + a X :^ + b, by adding ^ a' to both fides, we ilill 

 prelerve the equality, and render the firft fide a complete 

 fquare, that is, we have 



x' ± ax + ^a' — :J« + b ; 

 and, by extraction, 



" ±1" = ± V ii"' ± i) ' 

 confequently 



x=z + -^± ^{^a' + b), 



which form includes all the preceding formuls. We ftiali 

 not here give the folution of any examples, as the reader will 

 find fufficient exercifes under the article Equation ; but it 

 may not be amifs to Ihew other methods of obtaining the 

 roots of quadratic equations, viz. by fines and tangents, con- 

 tinued fractions, &c. 



T'je Solution of Quadratic Equations by tjieans of a Table <yf 

 Sines, Tangents, iSfc. — Here, referring to our preceding four 

 forms, the folution of them may be exhibited as follows ; 



Form I. X' -\- a X ^^ ~ b. 



Put - ^/ A 

 a 



fin. 2 ; then 



= {=: 



- r 



— ^ b X tan. 

 /by. cot. 



I:} 



fl - cof. a)1 

 (I + cof. 2)1 



or. 



1-. ka (i + cof. 2). 

 Form 2. .v' — a X z= — b. 



^/ i =: fin. z ; then 



+ a/* X 



Put- 

 a 



tan. i 

 -f V ^ X cot i 



= { 



f^} 



- cof. 2) \ 

 + cof. 2} j 



Form 3. x' + a X = b. 

 2 

 Put - ^/ A = tan. 2 



then 



^ _ ( + \/ ^ X tan. 1=7 



*- I- ,/A X cot. |2i 



_ f -I- -i a ( fee. 2 — I ) 7 



•^- I- l^(fec. 2 + I); 



Form 4. x' 

 „ 2 

 Put - V 



(fee. 

 i3 X = b. 

 = tan. 2 ; then 



-{'-'. 



b X cot. ■§27 

 b X tan. 1=5 



_ f -)- ia (fee. 2 4-1)7 



The above formulse refult immediately from the co:>- 

 ilruftion of quadratic equations. Thus, in the two finl 

 forms, let A E, {Plate XIII. fg. 6. Anahfis,) reprefent a, 

 and C D -= 'A, then A D and D B will be the required 

 roots from the known conltruAion of equations. See Con- 

 struction. 



Now here it ie obvious that the fin. % reprefents the fin. of 



CED, 



