The above solution is generally known as Cardan’s Solution, as it was first published by him in the Ars Magna, in 1545. Cardan obtained the solution from Tartaglia; but the solution of the cubic seems to have been due originally to Scipio Ferreo, about 1505.
In this solution we assume x=y +z, and from (2) find z=-{q}{3y} hence to solve a cubic equation of the form
x^3+ qx+r=0
we substitute y - {q}{3y} for x.
Ex. Solve the equation x^3 - 15x = 126.
Put y - -15 3y = or y + 5 for a, then whence y^6 — 126 y^3 = — 125. y^3= 125, y = 5.
But x=y+ 5 y = 6.
Dividing the given equation x3—15x—126=0 by x—6, we obtain the depressed equation x2+6x+21=0, the roots of which are — 3+ 2 —3, and -3-2-3.
Thus the roots of x^3 - 15x = 126 are 6, —3+2-3, and —3—2—3.
BIQUADRATIC EQUATIONS.
621. We shall now give a brief discussion of some of the methods which are employed to obtain the general solution of a biquadratic equation. It will be found that in each of the methods we have first to solve an auxiliary cubic equation; and thus it will be seen that as in the case of the cubic, the general solution is not adapted for writing down the solution of a given numerical equation.
