1. Cubic Equations 7 example of such an equation with a quadratic term: x3 - 3x2 - 3x - 1 = 0. To solve such equations, Cardano transformed them using a generally applicable procedure into equations of the form y3 + py + q = 0. Starting with a cubic equation in the general form x3 + ax2 + bx + c = 0, the transformation consists in adding the summand | to the desired solution x, which allows the quadratic and cubic terms to be com- bined: 9 / a\3 a2 a3 / a\3 a2 / a\ 2 o + aX =(X+3) - J * - 2 7 = ( * + 3 ) - j ( * + 3 ) + ^ a - x3 To obtain the complete transformation of the coefficients of this equation, one replaces every occurrence of x in the equation via the substitution a x = y , y 3 ' obtaining, after collecting terms in like powers of y, the identity x3 + ax2 + bx + c = y3 + py + q, with P = ~ 3 a 2 + b ' 2 3 ! L Once one has solved the reduced cubic equation y3 + py + q = 0 with Cardano's formula, the solution of the original equation can be obtained with the transformation x = y | . In the concrete example x3 3x2 3x 1 = 0, the transformation x = y + 1 leads to the equation ^ - 6j/ - 6 = 0, whose solution y = ^ 2 + ^ 4 obtained from Cardano's formula leads to the following solution of the original equation: x = 1 + v^2 + v7!.
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