Where it is evident, by inspection, that z=1.

And if this number be substituted for r, 0 for b, and 17 for d in the two quadratic equations in the above rule, their solution will give

-

Which are the four roots of the proposed equation; the two first being real, and the two last imaginary.

b

Hence, also, since s+q=a+p2, and s―q——, there will arise, by addition and subtraction,

where p being known, s and q are likewise known.

And, consequently, by extracting the roots of the two assumed quadratics x2+px+9=0, and x2+rx+s=0, or its equal x -px +s=0, we shall have

-

x=—§ƒ±√(‡p2 −q); x=}p±√({p2 −s);

which expressions, when taken in + and -, give the four roots of the proposed biquadratic, as was required.

Where it may be observed, that when p, in the above cubic equation, is rational, the question may be solved by quadratics.

4. Given x48x3+14x2+4x=8, to find the four

roots, or values of x.

13+✓✓5,3-√5 Ans. 1+√3, 1-√3

5. Given x4-172-20x-6=0, to find the four roots,

or values of x.

6. Given - 27x3 +162x2+356x-1200-0, to find

the four roots of the equation.

2.05608, -3.00000

13.15306, 14.79086

7. Given x4 - 12x2 + 12x-3=0, to find the four roots

of the equation.

RESOLUTION OF EQUATIONS.

BY APPROXIMATION.

EQUATIONS of the fifth power, and those of higher dimensions, cannot be resolved by any rule or algebraic formula, that has yet been discovered; except in some particular cases, where certain relations subsist between the coefficients of their several terms, or when the roots are rational; and, for that reason, can be easily found by means of a few trials.

In these cases, therefore, recourse must be had to some of the usual methods of approximation; among

N

which that commonly employed is the following, which is universally applicable to all kinds of numeral equations, whatever may be the number of their dimensions, and though not strictly accurate, will give the value of the root sought to any required degree of exactness.

RULE.

Find, by trials, a number nearly equal to the root sought, which call r; and let z be made to denote the difference between this assumed root, and the true root x.

Then, instead of x, in the given equation, substitute its equal r±z, and there will arise a new equation, involving only z and known quantities.

Reject all the terms of this equation in which z is of two or more dimensions; and the approximate value of z may then be determined by means of a simple equa

tion.

And if the value, thus found, be added to, or subtracted from that of r, according asr was assumed too little, or too great, it will give a near value of the root required.

But as this approximation will seldom be sufficiently exact, the operation must be repeated, by substituting the number thus found, for r, in the abridged equation exhibiting the value of z; when a second correction of z will be obtained, which, being added to, or subtracted from r, will give a nearer value of the root than the former.

And by again substituting this last number for r, in the above mentioned equation, and repeating the same process as often as may be thought necessary, a value of may be found to any degree of accuracy required.

Note. The decimal part of the root, as found both by this and the next rule, will, in general, about double itself at each operation; and therefore it would be useless,

as well as troublesome, to use a much greater number of figures than these in the several substitutions for the values of r. (i)

EXAMPLES.

1. Given x3x2+x=90, to find the value of x by approximation.

Here the root, as found by a few trials, is nearly equal to 4.

Let therefore 4=r, and r+z=x.

x3=r3+3r2z+3rz2+z3

Then rr2+2rz+z2

x=r + z

=90.

And by rejecting the terms 23, 3rz2 and z3, as small in comparison with 2, we shall have

r3+r2+r+3r2 z+2rz+z=90;

(i) It may here be observed, that if any of the roots of an equation be whole numbers, they may be determined by substituting 1, 2, 3, 4, &c. successively, both in plus and in minus, for the unknown quantity, till a résult is obtained equal to that in the question; when those that are found to succeed, will be the roots required.

Ör, since the last term of any equation is always equal to the continued product of all its roots, the number of these trials may be generally diminished, by finding all the divisors of that terni, and then substituting them both in plus and minus, as before, for the unknown quantity, when those that give the proper result will be the rational roots sought: but if none of them are found to succeed, it may be concluded that the equation cannot be resolved by this method; the roots, in that case, being either irrational or imaginary.

Again, if 4.1 be substituted in the place of r, in the last equation, we shall have

90-r3-2 — z
3r2+2r+1

90-68.921-16.81-4.1

=.00283

50.43+8.2+1

And consequently x=4.1+.00283=4.10283, for a second approximation.

And, if the first four figures, 4.102, of this number be again substituted for r, in the same equation, a still nearer value of the root will be obtained; and so on, as far as may be thought necessary.

2. Given approximation.

2+20=100, to find the value of x by Ans. x 4.1421356

3. Given 3+9x2+4x=80, to find the value of x by approximation. Ans. x 2.4721359

4. Given x4—38x3+210x2+538x+289=0, to find the value of x by approximation.

Ans. x=30.535€5375

5. Given +64 — 10x3-112x2 - 207x find the value of x by approximation

Ans. 4.46410161

The roots of equations, of all orders, can also be determined, to any degree of exactness, by means of the following easy rule of double position; which, though it has not been generally employed for this purpose, will be found, in some respects, superior to the former, as it can be applied, at once, to any unreduced equation, consisting of surds, or compound quantities, as readily as if it had been brought to its usual form.

RULE.

Find, by trial, two numbers as near the true root as possible, and substitute them in the given equation in'stead of the unknown quantity, noting the results that are obtained from each.