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10. Given x3 – 22x=24, to find the root of the equation, or the value of x.

Ans. I=5.162277 11. Given x3 - 1722 +54x=350, to find the root of the equation, or the value of x.

Ans. x=14.954068


A biquadratic equation, as before observed, is one that rises to the fourth power, or which is of the general form

x 4 tax3 +bx+ca+d=0. Or, when its second term is taken away, of the form

x4 +bx+cx+d=0, To which it can always be reduced ; and in that case, its solution may be obtained by the following rule : Find the value of 2 in the cubic equation 1

1 1 1
-63 +--C2

and let the root thus determined be denoted by r.

Then find the two values of x, in each of the following quadratic equations.

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be + d)2=108

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x++(v{?(r-}6}}}x=-(++v{(r+%632 – d}

-(v{{(r=}}}}x==(1+0)-v{(r+%) –d}

and they will be the four roots of the biquadratic equation required. (h)

(h) The method of solving biquadratic equations was first dis. covered by Louis Ferrari, a disciple of the celebrated Cardan, be

Or the four roots of the given equation, in this last case, will be as follows :

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1. Given x4 +12x – 17=0, to find the four roots of the equation.

Here a=0,b=0, c=12, and d=-17;

fore mentioned; but the above rule is derived from that given by Descartes in his Geometry, published in 1637, the truth of which may be shown as follows : Let the given or proposed equation be

34 + ax2 + bx topo and conceive it to be produced by the multiplication of the two quadratics

*3 +px+9=0, and x2 trx ts=0. Then, since these equations, as well as the given one, are each =0, there will arise, by taking their product, *44(tr)x3 +(8+9+pr)x3 +(p8tar)x+98=x4 +ax2 +bx+c.

And, consequently, by equating the homologous terms of this last equation, we shall have the four following equations, pt=0; statpr=d; pst greb; 98=sc. ·

or r=-p;s+q=a + p3, s-q=-, q8=C.

p Whence, subtracting the square of the third of these from that of the second, and then changing the sides of the equation, we shall have

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Whence, by substituting these numbers in the cubic equation


1 1 23

-63+ -ca

108 8 we shall have, after simplifying the results,

23 +172=18, Where it is evident, by inspection, that z=1. And if this number be substituted for r, o for b, and

17 for d in the two quadratic equations in the above rule, their solution will give

-1/2+v-itv18=-i2+v-}+3.72 x=-1/2-V-+18=-1/2-v-}+3,2

= V2V x=+v2+v-}-V18=+iv2+v-1-32

x=+iv2--1-18=+iv2-v-1-3.72 Which are the four roots of the proposed equation ; the two first being real, and the two last imaginary.





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a2 +Zupa +p4_62

=498, or 4c; or po +2ap4+(42 – 4c)p2 =b . Where the value of p may be found by the rule before given for cubic equations. Hence, also, since 8+9=u+p2, and s957

there will arise, by addition and subtraction,


b 1 202

+ 2

;9 a t-p2 2p

2p where p being known, s and are likewise known.

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

2 =-p+V (#p2-9); x=spěv (pa -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.

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2. Given 24 -55x2 – 30x +-50450, to find the four roots, or values of x.

Ang. 3, 7,-4, and -6 3. Given x4 +-2x3 – 7,002 - 8x=-12, to find the four roots, or values of x.

Aps. 1, 2, -3, and -&. 4. Given x4 -8x3 + 14x2 + 4x=8, to find the four roots, or values of x.


3+5, 3-5


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5. Given x4 -17? — 20x—6=, to find the four roots, or values of x.


2+17, 2-77

—2+12, -2-12 6. Given 24 - 27x3 +162:32 +356x – 12000, to find the four roots of the equation.

2.05608, -3.00000


Ans: { 13.15508, 14.79036

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


.606018, -- 3.907378 2.858083, .443277





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


which that commonly employed in 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 valueof the root sought to any required degree of exactness.



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 rIz, 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 equation.

And if the value, thus found, *be added to, or subtracted from that of r, according as r 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 x 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,

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