therefore, to show, that every complete equation of this nature may be transformed into another, where the second term is wanting, and, therefore, that the rule is in fact applicable to all cases.

In order to prove this, let us take the complete equation, x3-6x2+11x-6=0;

Here, if we take the third of the coefficient 6 of the second term, and make x-2=y, we shall have

We have therefore the equation y-y=0, the resolution of which is evident, since it is the product of the factors, y × (y+1) × (y−1)=0.

445. In general, therefore, let there be given the complete equation,

x2+ax2+ bx+c=0,

of which it is required to destroy the second term.

For this purpose we must join to a third of the coefficient of the second term, preserving the sign, and for this sum substitute a new quantity, as y, so that x+

1

1

=y; from

whence we have x=y- a; whence results the following cal

3

culation:

446. Let it, therefore, be proposed to resolve the equation, x36x2+13x-12=0.

Assume xy+2, we shall then have

x3y3+6y2+12y+8,

- 6 x2 — — 6 y2 — 24y-24,

which gives y3+y-2=0. Hence we have ƒ=-1 and g=2.

3/27-6√21= 21 = 1 ( 3 + √ 21 ) + ( 3 = √21) = 1 + 1 = 1.

3

2

2 2

1 3

Hence

Hence xy+2=3, one of the roots of the equation proposed, from which the others may be easily found.

In the above example it happened that the binomials 27+6/21 and 27-6/21, were in fact real cubes, and that we consequently were enabled to extract the cube roots, and destroy the radical sign; which can only be the case when the equation has a rational root, and therefore the rules given in the preceding chapter are more easily employed for finding that root. On the other hand, when there is no rational root, it is impossible to express the root which we obtain in any other way than according to the rule of Cardan. For instance, in the equation,

x3=6x+4,

we have ƒ 6 and g=4, so that

x=/ (2+2√− 1) + 3⁄4/ (2−2 √1)

which cannot be otherwise expressed.

CHAP. LV.

OF THE RESOLUTION OF EQUATIONS OF THE FOURTH

DEGREE.

447. THE general form of an equation of the fourth degree, or a Biquadratic, is

x2+ax3 + bx2+cx+d=0.

We shall in the first place, however, consider pure equations of the fourth degree, the expression for which is simply x=f; the root of which is immediately found by extracting the biquadratic root of both sides, whence we obtain x=√ƒ.

As a is the square of x, the calculation is greatly facilitated by beginning with the extraction of the square root, for

we shall then have af. And taking the square root again, we have r√. For example, in the equation 2*2401, we have first x2=49, and secondly x=7.

It is true that this is apparently but one root, whereas an equation of the fourth degree must have four roots; but the method that we have explained will be found actually to give four roots; for in the above example we have not only x2=49, but also x2-49; now the first value gives x=7 or -7, and the second value gives x=-49=7—1, and also x=-49-7-1, which are the four biquadrate roots of 2401. The same is true of all other numbers.

Next to these pure equations, we shall consider others in which the second and fourth terms are wanting, and which have the form,

x++fx2+g=0.

These, as we have already seen (Art. 223.), may always be considered as quadratic equations, and resolved by any of the methods given in Chapter XXIV. For we may either make a2=y, whence we shall have

or, proceeding at once to complete the square in the original equation, we shall have,

as before, where the double signs ± indicate all the four

roots.

448. But when the equation contains all the terms, it may then be considered as the product of four factors. In fact, if we multiply these four factors together, (x−p) × (x−q) × (x −r) × (x−s), we obtain the product xa— (p+q+r+s)x3 + (pq+pr+ps+qr+qs+rs) x2 — (pqr+pqs+prs+qrs)x+pqrs, and this quantity cannot be equal to 0, except when one of these four factors is 0. Now this may happen in four

ways:

3. When =r;

1. When x=p;

4. When xs;

2. When x=q;

and consequently these are the four roots of the equation.

449. If we consider this formula with attention, we shall find in the second term the sum of the four roots multiplied by; in the third term the sum of all the possible products of two roots multiplied by 2; in the fourth term the sum of the products of the roots taken three and three, multiplied by ; and lastly, in the fifth term the product of all the four roots multiplied together.

450. Now, since the last term contains the product of all the roots, it is evident that such an equation of the fourth degree can have no rational root, which is not a divisor of the last term. This principle, therefore, furnishes an easy method of determining all the rational roots, when there are any; since we know that they must be found among the divisors of the last term; and having found one such root, as x=p, we have only to divide the equation by x-p, by which means we shall obtain an equation of the third degree, which may be resolved by the rules already given.

For this purpose, however, it is necessary that all the terms should consist of integers, and that the first should have only unity for the coefficient. Whenever, therefore, any of the