Effection. 1. Bring the Abfolute Number to one fide of the Equation (In. 432.) 2. Add the Square of the Coefficient of the fecond Term to both fides of the Equation. 3. Extract the Square Root from both fides. 4. Add or fubtract one half the faid Coefficient to or from both fides, and it is done. Example 1. Refolve the Equation a-4a77=0, or a2+4a=77 into its conftituent Roots, which are one Pofitive, and the other Defective. Example 3. Refolve a2+32a-255-0, or a2-32a+255=0. 507. To extract the Root from an Adfected Cubic Equation by Approximation, according to In. 354. Example 1. Let it be required to extract the Root from the Adfected Cubic Equation a3-28.25a2-91.75a-62.5=0; which confifts of one Pofitive and two Defective Roots (In. 465.) And the Pofitive Root is fome Number between 30 and 40 (In. soi.) therefore make b=30 less than just, i. e. 30+e or b + c = a +a2=+27000+2700e+9e2+e -28.25a2——25425 -1695e-28.2562 1240 +913.25e+61.75e*+e3=0 Then Then putting d=1240, 5=913.25, 161.75, we have the Equation -d +se+tee+eee=0, or, I jse+tee+eee=d 2 seteed rejecting e Or 2-se 3 d-se ee= 4 t de-see Whence b+ 31.25007 a true to the feventh Figure at the firft Oper ation, or a 31.25 just; then a3-28.25a-91.75a–62.5=0 a―31.25=0 -=a2+za+2=0 And the two Roots of a+za+2=0 are -1 and -2. Therefore all the Roots of the given Equation are +31.25, -1 and -2. Example 2. Refolve the Equation a3-6a2+24a-20039=0. The greatest Pofitive Root in this Equation is between 20 and 30 (In. 501.) but nearer the latter, therefore affume 30=b more than just, i. . ba + a3=+27000-2700e+90ee-eee 6a2 5400+ 360e- bee +249 + 720- 240 -20039-20029 b-10 tt-s 1|b-e=28.999994 a true to the eighth Figure at the first Operation, or a= 29 exact for the firft Root Pofitive. Then a3—6a2+24a—20039=0=a2+23a+691=0, whence 23 4 2lw2 3-11.5 a-29-0 1|a2+23a=—691 2 a2+23a+132.25=+132.25-691=-558.75 4a=+-558.75 -11.5 the fecond Root which is impoffible (In. 470.) And the Reason you fee is, because the Square of one half the Coefficient of the second Term, i. e. the Square of 22 is lefs than the Abfolute Number 691. 23 Lastly, a2+23a+691 a――558.75+11.5 the third Root impoffible.. =a+5+1. Whence a=— SCHOLIUM XII. 508. In the foregoing Examples I have purposely omitted affuming any more than the firft Figure of the Root; but if more can be affumed, the first Operation will feldom or never miss of bringing forth quintuple the Figures at least, but generally more, as may be feen. But this Method fails in all Adfected Equations of above three Dimenfions: And therefore for the Refolution of thefe, we must have Recourse to the following Method of Dr. Hally, which, I believe, by far exceeds all the Methods that ever have been hitherto invented for the like Purpose. PROBLEM XVII. 509. To extract the Roots from all kind of Equations by Approximation. Example 1. Let it be required to refolve the Equation-3330a2-1600a-8125=0. The greatest Pofitive Root of this Equation is fome Number between 320 and 330 (In. 502.) therefore affume b=320 lefs than juft, i. e. 320+e=a |