Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 1, A-AME

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(!e9 tlle origin*1 equation, for
—- —x 2— —6.
12X12
6
-X2~l2, alfo
Fx. 3. Given + 28=iia', to find x.
Then a?*—11^=: —28.
And, completing the fquare, «*—lltf-t-
4 4
-28=9.
4
IX 5
Therefore, by extradling the root, x— — -—
Hence xz= — —that is, x= -f- 7, or x— -{-4.
In the firft two examples, we found onepofitive value
for x in each, and alfo one negative value 5 but in this
example both the values of x are politive, and, upon
trial, each of them is found to fatisfy the equation 5 for
JX7 + 28=iiX7, alfo 4 X 4 + 28=11 X 4»
149. As at firll fight it appears remarkable, that in
every quadratic equation the unknown quantity admits
always of two diftindl values, or roots, it will be proper
to confider a little farther the circumftances upon which
this peculiarity depends. This is the more neceflarv, as
the property of the unknown quantity admitting of feve-
ral values is not peculiar to quadratics, but takes place
alfo in equations of the higher degrees, where tliecaufe
of the ambiguity requires an explanation fomewhat dif¬
ferent from that which we have already given in the
prefent cafe.
150. Let us again confider the equation xa + 2x—
35, which forms the firft of the three preceding exam¬
ples ; by tranfpoling all the terms to one fide, the fame
equation may be alfo expreffed thus, x* + 2x—35=05
£0 that we fliall have determined x, when we have
found fuch a number, as when fubflituted for it in the
quantity x* + 2x—35, will render the refult equal to o.
But x2-J-2x—35 is the product of thefe two factors
x—5> ant^ x+_7'> as may be proved by actual multipli¬
cation 5 therefore, to find #, we have (a?—5) (at4-7)
=0 5 and as a product can only become =0, when
one of its fadtors is reduced to o, it follows, that either
of the two fadtors x—5 and #-|-7 may be affumed
=05 if x—5=0, then *=5; but if a?4-7=0, then
x——7, fo that the two values of at, or two roots of
the equation xi-\-2x=7)^ are 4“5 and —7, as we have
already found in.a different manner.
151* What has been juft now fhewn in a particular
cafe is true of any quadratic equation whatever, that
is, if x* srpx=q, or by bringing all the terms to one
fide, x*+px—y=o, it is always poffible to find two
fadtors x-j-w, and x—b, fuch, that x*-\-px—q=(x-\-a')
(x—b), where a and b are known quantities, which
depend only upon p and q the given numbers in the
equation, and fmee that to have (x—a) (x-f£)=o, we
may , either affume x—a=o, or x 4-^=0, it evidently
follows, that the conditions of the equation x* +px—q
=0, or x% + px=q are alike fatisfied, by taking a?=4-«
or x=—b.
Irom thefe confiderations, it follows, that x can
have only two values in a quadratic equation 5 for if
it could be fuppoled to have three or more values,
2
E B It, A. 529
then it would be pollible to refolve x^ into as Quadratic
many fadtors, *—c, x—(J, &e. 5 but the product of Equations,
more than two fadtors muft neceffarily contain the thirdv—-v——J
or higher powers of x; and as x1-\-px—q contains no
higher power than the fecond j therefore no fuch refo-
lution can take place.
152. Since it appears that x*+px—q may be confi-
dered as the produdt of two fadtors at—a, and x-\-b,
let us examine the nature of thefe fadtors 5 according¬
ly, taking their produdt by adtual multiplication, we
find it a;“4"(^—a)x—ab; and fince this quantity muft
be equal to x'+px—7, it follows, that b—a=p and
ab=q? or, changing the figns of the terms of both
equations, a—b=—-77, —ab=—q. Now if we confi¬
der that 4-0, and —are the roots of the equation
x1 +px=q ; it is evident that a—b is the fum of the
roots, and —ab their produdt. So that from the equa¬
tions a—b=—77, and —ab=q, we derive the follow-
ing propofition relating to the roots of any quadratic
equation. 4 he fum of the roots of any quadratic
equation a?1 4-^=7 isr equal to —77, that is, to the co¬
efficient of the fecond term, having its fign changed 5
and their produdt is equal to —7, or to the latter fide
of the equation, having its fign alfo changed.
153- This propofition enables us to refolve feveral
important queftions concerning the roots of a quadra¬
tic equation, without adtually refolving that equation.
Thus we learn from it, that if 7, the term which does
not involve the unknown quantity, (called fometimes
the abfolute number) be pofitive, the equation has one
of its roots pofitive, and the other negative 5 but if that
term be negative, the roots are either both pofitive or
both negative. It alfo follows, that in the former cafe
the root which is denoted by the leaft number will have
the fame fign with the fecond term, and in the latter-
cafe, the common fign of the r-oots will be the contrary
to that of the fecond term.
154. From this property of the roots we may alfo de¬
rive a general folution to any quadratic equation x1+px
=7; for vve have only to determine two quantities
whofe fum is —p, and produdt —7, and whole quanti¬
ties fliall be the two values of x, or the two roots of the
equation.
Without confidering the figns of the roots, let us call
them v and 2;, then
v + %=—77, and v%=—7. ,
From the fquare of each fide of the firft equation let
four times the fecond be fubtradted, and we have
V1 2VZ + Zt=p2-^-^q, or (v—2;)* =77* 4-47,
therefore by extradting the fquare root, v—25=rt:
/Jp'+Atq i from this equation, and from the equation
v-\-‘Z=p, we readily obtain v=—
-p=t= ‘Jp2+i\q
%=■
-, that is, if v=-
-P+ ^ Pz +
4?
then 2;:
2
then ^=±-^+47
a„d if v=-=+l+Wt
But the value of upon the one fuppefition, is the
fame as the value of 2; upon the other fuppofition, and
vice verfa; therefore, in reality, the only two diflindl
values