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Then substitute 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 proposition an equation, in which the coefficient of the firit term is any known quantity, as a, may be transformed into another, in which the coefficient of the first term shall be unit. Thus, let the equation be ax?—,5x2 + x-r=o. Suppose y=ax, or x=

x=ž, and for x and its powers insert , and its powers, and the équation becomes +%-r=o, or y -pya+qay—a?r=o. Also, let the equation be 583—6x2 +71—30=0; and if

then 93-6y2 +359—750=0. Cor. 2. If the two transformations in Prop. 2. and 3. be both required, they may be performed either separately or together.

Thus, if it is required to transform the equation ax:--px? +qxấro into one which thall want the second term, and in which the coefficient of the first term shall

be

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a?

у

x=

а

be 1; let x=2, and then yo—pj2+qay~aér=0 as before ; then let y=xtip, and the new equation, of which z is the unknown quantity, will want the second term; and the coefficient of zo, the highest term is 1. Or, if x=2+1, the same equation as the last found will arise from one operation.

Ex. Let the equation be 5x2–6x2+2x– 30 =0. If x=, then y—6y* +358– 750=o. And if y=x+2, 7+232-696 30. Also, at once, let x=?, and the equation properly reduced, by bringing all the terms to a common denominator, and then casting it off, will be gi+232–696

5

=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 surd coefficient be taken away in certain cases.

Cor,

Cor.

4. Hence also, if the coefficient of the second term of a cubic equation is not: divisible by 3, the fractions thence arising in the transformed equation, wanting the second term, may be taken away by the preceding corollary. But the second term also may be taken

away,

so that there shall be no such fractions in the transformed

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3 , +p being

equation, by suppofing x=
the coefficient of the second term of the gi-
ven equation. And, if the equation ax3
-px+qx-r=o be given, in which p is

2+P
not divisible by 3, by supposing x=
the transformed equation reduced is %
3p? +919X2—2p+gapa--27a?r=0;
wanting the second term, having 1 for the
coefficient of the first term, and the coeffi-
cients of the other terms being all integers,
the coefficients of the given equation being
also supposed integers.

General Corollary to Prop. I. II. III.

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

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roots

2+2
5

* the

roots of the original equation, from which they were derived, will easily be found from the simple equations expreffing their relation. Thus, if 8 is found to be a root of the transformed equation z'+2324696 =0, (Cor. 2. prop. 3.), since x = corresponding root of the given equation,

8+2 5x-6x2 +7*—30=0, must be

= 2. It is to be observed also, 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 cubics.

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"ROM the preceding principles and o

perations, rules may be derived for resolving equations of all orders.

FRO

I. CARDAN's Rule for Cubic Equations.

1

The second term of a cubic equation being taken away, and the coefficient of the first term being made i, (by Cor. 1. Prop. 2. and Cor. 1. Prop. 3. Chap. II.) it may be generally represented by *** +39x+2r =0; the sign + in all terms denoting the addition of them, with their proper signs. Let x=m+n, and also mn = -9; by the fubftitution of these values, an equation of

the

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