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Pi

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or, if we replace 24-p, by p and — 21+ - p, by q

we get

2p

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y3 = py + q,

which is the form required in equation (c) where x takes the place

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Hence we may assume that the given cubic has the reduced form

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where x equals the sum of the positive real cube roots of s and t. From (4) we have

3

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3

=

33√st · x + s + t,

replacing 'Vs+Vt by x; but if the values of x and of x3 are the same in equations (4) and (5), then

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s+ t = q.

Hence s and t are the roots of the quadratic equation (422)

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In order to obtain in this way the values of the three roots of the given equation (c), we put for the real values of the cube roots,

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If 2, i. e. q2

4

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p3, is positive, which is always the case when p in equation (c) is negative, then the equation x3 + px±q = 0 has one real and two imaginary roots. *

If 0, then we have three real roots, two of which are equal; one of the equal roots must have one-half the value of the third root, with the opposite sign.

If is negative, r is imaginary, and all of the roots according to Cardan's formula appear to be imaginary, but in fact in this case all of the roots are real. This is called the irreducible case, because we do not yet have the means to reduce the imaginary form to a real form.

* Cardan's formula does not often give the rational root of a cubic, if it has a rational root, in a rational form. In order that this may happen, it is necessary, according to a discussion due to E. Liebknecht, that the cubic equation shall have the form

x3 = 3 mnx + m3 + n3,

where m and n are arbitrary rational numbers. Cardan's formula then gives x = m + n.

EXERCISE CXIII

The following equations have at least one rational root and are of such a kind that Cardan's formula gives this root in rational form. Find all the roots of the equations by means of Cardan's solution:

1. x3 3x + 2.

3. x3 = 9x- 28.

5. x3 18x=35.

2. x3 = 36x + 91.

4. x3+9x+ 26 = 0.
x3 72x 280= 0.

6.

The following equations have one real rational or irrational root. Cardan's solution gives also the rational root in an irrational form. In this case the square and cube roots must be found, since frequently the irrational form can only be brought to a rational form by complicated calculations.

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The following equations have likewise one real rational or irrational root, but must first be more or less transformed in order that the Cardan solution can be applied to them:

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19. 3x+13x2+11x140. 20. 28.3-126x2+195x-139=0.

B.

THE TRIGONOMETRIC SOLUTION

773. The equation

(c)

x3 = px + q

may be solved in the following manner, by Trigonometry, in the irreducible case, when q3 is negative.

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Equate the coefficients of the two equations; the result is

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where

must be found by aid of the trigonometric tables. Having found 6, the three roots of the equation are given by the equations:

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The roots of the equation ax3

= px q are the same as those in (2) with opposite signs. It follows from the solution of the equation

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not belong to the class of cubics discussed here, since if Р is negative then q3 is positive, it belongs to the Cardan solution; but the class of equations discussed requires that q2 — „p3 < 0.

4 27

4 27

The following cubic equations have three real roots, some of them rational and some irrational. It is required to solve the equations by the trigonometric formulae in (2). If the equation is not in the given reduced form, then it must first be reduced to this form.

EXAMPLE.-Solve the equation x37x5.

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774. In solving a cubic equation, one must determine first whether the cubic has one rational root or not. If the equation has a rational root it can be readily solved by 8771; if it does not have a rational root, it must be reduced to the form x3 = px + q by 772, in case it does not already have this form.

Then the sign of

must be determined.

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If this sign is +, then Cardan's solution is to be applied; if this sign is, the trigonometric solution must be

applied.

C. TRIGONOMETRIC SOLUTION OF CUBIC EQUATIONS WITH TWO IMAGINARY ROOTS

775. The case in which one only of the roots of the cubic is real can be handled trigonometrically. This will be the case when the cubic is written in the form

and

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is positive under all conditions; or in case the cubic is written in the form

where

x3 — 3 lx + 2 q = 0,

(2)

73 < q2; hence v l3÷q<1.

[8772, (8) fr.]

3

In the first case put cos 2P =

7 and tan P=3 tan 6, then V q2 + B

after trigonometric transformations of the values found by Cardan's formula, 772, we obtain the following solutions of equation (1):

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x=(~sin2P

1

sin 2P

and tan P'tan 0,

2

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VI, sin 2P

1

sin 2P

+ i√3 cot 2P)√T, x=(-2-i√3 cot 2 P)√ī.

EXERCISE CXIV

Test whether the following equations have one or three real roots and solve the equations by the trigonometric method which is applicable.

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