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Then fubftitute for x and its powers,

or ye and its powers, and the new equation of which y is the unknown, quantity, will have the property required.

Cor. 1. By this propofition an equation, in which the coefficient of the firft term is any known quantity, as a, may be tranfformed into another, in which the coefficient of the first term fhall be unit. Thus, let the equation be ax-x2+qx-r=0; Suppofe y=ax, or x= x=2, and for x and


its powers infert 2 and its powers, and the

equation becomes


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—py2+qay—a2ro. Alfo, let the equation be 5x3-6x2+7x—30=0; and if x=3, then y3—6y2+353—750=0.


Cor. 2. If the two transformations in Prop. 2. and 3. be both required, they may be performed either feparately or together.

Thus, if it is required to transform the equation ax3-px2+qx-r-o into one which fhall want the fecond term, and in which the coefficient of the firft term fhall


be 1; let x=2, and then y3—py2+qay—


a2r=Q as before; then let y=x+1⁄2p, and the new equation, of which z is the unknown quantity, will want the second term, and the coefficient of x3, the highest term is 1. Or, if x =*+, the same equation



as the last found will arife from one operation.

Ex. Let the equation be 5x3-6x2+7x— 30=0. If x -23, then y3—6y2+351— 7500. And if y=x+2, x3+23≈—696

o. Alfo, at once, let x=


and the

equation properly reduced, by bringing all the terms to a common denominator, and then cafting it off, will be x3+23%-696 o, as before.

Cor. 3. If there are fractions in an equation, they may be taken away, by multiplying the equation by the denominators, and by this Prop. the equation may then be transformed into another, without fractions, in which the coefficient of the first term is 1. In like manner, may a furd coefficient be taken away in certain cafes.



4. Hence alfo, if the coefficient of the second term of a cubic equation is not divisible by 3, the fractions thence arifing: in the transformed equation, wanting the fecond term, may be taken away by the preceding corollary. But the fecond term alfo may be taken away, fo that there shall be no fuch fractions in the transformed

equation, by fuppofing x = +p being

3 ?

the coefficient of the fecond term of the given equation. And, if the equation ax3 -px2+qx-r=o be given, in which is

not divifible by 3, by fuppofing x=



the transformed equation reduced is z3· 3p2 +9aq xx-2p+9apq—27a2r=0; wanting the second term, having 1 for the coefficient of the firft term, and the coefficients of the other terms being all integers, the coefficients of the given equation being alfo fuppofed integers.

General Corollary to Prop. I. II. III.

If the roots of any of thefe transformed equations be found by any method, the

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roots of the original equation, from which they were derived, will eafily be found from the simple equations expreffing their relation. Thus, if 8 is found to be a root of the transformed equation 3+23%-696

=0, (Cor. 2. prop. 3.), fince x=


+2, the

correfponding root of the given equation, 5x —6x2+7x—30—0, must be3+



=2. It

is to be obferved alfo, that the reasoning in Prop. 2. and 3. and the corollaries, may be extended to any order of equations, though in them it is applied chiefly to cu




Of the Refolution of Equations.



ROM the preceding principles and operations, rules may be derived for refolving equations of all orders.

I. CARDAN'S Rule for Cubic Equations.


The fecond term of a cubic equation being taken away, and the coefficient of the first term being made 1, (by Cor. 1. Prop. 2. and Cor. 1. Prop. 3. Chap. II.) it may be generally reprefented by +39x+2r =0; the sign + in all terms denoting the addition of them, with their proper figns. Let x=m+n, and alfo mnq; by the fubftitution of thefe values, an equation of


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