Page:Elementary algebra (1896).djvu/505

487 Rh CARDAN’S METHOD FOR THE SOLUTION OF CUBIC EQUATIONS.

619. The general type of a cubic equation is

x^3 + Px^2 + Qx+ R=0,

but as explained in Art. 585 this equation can be reduced to the simpler form x^3 + qx +r = 0, which we shall take as the standard form of a cubic equation.

620. We proceed to solve the equation x^3 + qx +r = 0.

Let x=y+z; then

and the given equation becomes

At present y, z are any two quantities subject to the condition that their sum is equal to one of the roots of the given equation; if we further suppose that they satisfy the equation 3yz+q=0, they are completely determinate. We thus obtain

Solving this equation,

Substituting in (1),

We obtain the value of x from the relation x= y+z; thus