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lefs than the roots of the given equation. by 1e, Cor. 4. Prop. 2. When xỏ the refult is the abfolute term of the given equation. When for x is inferted -I = e, the refult is the abfolute term of an equation whofe roots exceed the roots of the given equation by 1e. Hence, if the terms of the feries, 1, 0, -1, -2, &c. be inferted fucceffively for x, the refults will be the abfolute terms of fo many equations, of which the roots form an increafing arithmetical feries with the difference 1. But, as the commenfurate roots of thefe equations must be among the divifors of their abfolute terms, they muft alfo be among the arithmetical progreffions found by this rule. The roots of the given equation, therefore, are to be fought for among the terms of these progreffions which are divisors of the refult, upon the supposition of x=0, because that refult is its abfolute
It is plain that the progreffion must always be increafing, only it is to be obferved, that a decreafing feries, with the fign +, becomes increafing with the fign
Thus, in the preceding example, -4, -3, -29 is an increasing feries, of which -3 is to be tried, and it fucceeds.
If, from the fubftitution of three terms of the progreffion, 1, 0, -1, &c. there arife a number of arithmetical feriefes, by fubftituting more terms of that progression, some of the feriefes will break off, and, of course, fewer trials will be neceffary.
III. Examples of Questions producing the bigber Equations.
It is required to divide 161. between two perfons, fo that the cube of the one's fharè may exceed the cube of the other's by 386.
Let the greater fhare be x pounds,
By the queftion, x-16-x-386
And by Inv. 2x^—48x2+768x—4096=386
x= O; - 2241
x=-1; - 2650
2650 - ́1, 2, 5, 10, 25, 53.
Where 8, 9, 10, differ by i; therefore +9 is to be tried; and being inferted for x, The two fhares then
the equation is =d.
are 9 and 7 which fucceed. Since x=9;
x-9=0 is one of the fimple equations from which this cubic is produced; therefore x3-24x2+384x-2241 = x2-15x+ 249=0.
And the two roots of this quadratic are impoffible.
What two numbers are those, whofe product multiplied by the greater will produce 405, and their difference multiplied by the lefs 20?
Let the greater number be x, and the lefs y.
Mult. and tranf. y+-+40y2—4°53′+400=8.
This biquadratic refolved by divifors, gives y=5; and therefore x=9. Alfo,
This cubic equation has one pofitive incommenfurate root, viz. 1.114, &c. which may be found by the rule in the next fection, and two impoffible. The incommenfurate root y = 1,114, &c. gives x=19.067, &c. and thefe two answer the conditions very nearly.
The fum of the fquares of two numbers 208, and the fum of their cubes 2240 being given, to find them.
Let the greater be x+y, and the less x—y. Then x+y2+xy|2 = 2x2+2y2 = 208. Hence y2 104—x2.
Also x+y+x-y| = 2x3· ·6xy2=2240. Subftitute for y2 its value and 2x3+624x
This reduced gives x3-156x+560=0.
The roots of this equation are +10, +4,
If x 10 then
14. y=2, and the numbers fought are 12 and 8, which give the only juft folution. If x=4, then y2=88 and y=√88. The numbers fought are therefore 4+88 and 4—√88. The last is negative, but they answer the conditions. Laftly, if x=-14, then y2-92, hence y=92, is impoffible; but ftill the two numbers —14 +√−92, −14−√—92, being inferted, would answer the conditions. But it has been frequently obferved, that fuch folutions are both useless, and without meaning.
IV. Solution of Equations by Approximation.
By the former rules, the roots of equations when they are commenfurate may be obtained: Thefe, however, more rarely occur; and when they are incommenfurate, we can find only an approximate value of them, but to any degree of exactness required. There are various rules for this purpose;