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487
ALGEBRA
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487
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
