Voorbeeld 1
Je wilt de vergelijking $1001x^3-933x^2+279x-27=0$ oplossen.
De waarde van $d$ is niet nul dus je kunt geen $x$ buiten haakjes halen. Er is ook niet direct een voordehandliggende oplossing in beeld. Je GR geeft mogelijk ook niet direct uitsluitsel.
De formule van Cardano
Invullen van de coëfficiënten $a$ t/m $d$ in de formule van Cardano geeft:
$
\eqalign{
& p = - \frac{{10884}}
{{1002001}} \cr
& q = - \frac{{358720}}
{{1003003001}} \cr
& W = \frac{{144i\sqrt 3 }}
{{1002001}} \cr
& x = \frac{3}
{7} \cr}
$
Je kunt dan verder oplossen met ontbinden. Eerst maar 's de staartdeling:
$ \eqalign{ & x - \frac{3} {7}/1001x^3 - 933x^2 + 279x - 27\backslash 1001x^2 - 504x + 63 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline {1001x^3 - 429x^2 } \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 504x^2 + 279x - 27 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline { - 504x^2 + 216x} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,63x - 27 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline {63x - 27} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \cr} $Dat schiet al lekker op...
Je krijgt:
$ \eqalign{ & 1001x^3 - 933x^2 + 279x - 27 = 0 \cr & \left( {x - \frac{3} {7}} \right)\left( {1001x^2 - 504x + 63} \right) = 0 \cr & x = \frac{3} {7} \vee 1001x^2 - 504x + 63 = 0 \cr & ... \cr} $
Met de ABC-fomule:
$ \eqalign{ & x_{2,3} = \frac{{504 \pm \sqrt {\left( { - 504} \right)^2 - 4 \cdot 1001 \cdot 63} }} {{2 \cdot 1001}} = \frac{{504 \pm \sqrt {1764} }} {{2002}} \cr & x_2 = \frac{{504 - \sqrt {1764} }} {{2002}} = \frac{3} {{13}} \cr & x_3 = \frac{{504 + \sqrt {1764} }} {{2002}} = \frac{3} {{11}} \cr} $
Oplossing
$ \eqalign{ & 1001x^3 - 933x^2 + 279x - 27 = 0 \cr & \left( {x - \frac{3} {7}} \right)\left( {1001x^2 - 504x + 63} \right) = 0 \cr & x = \frac{3} {7} \vee 1001x^2 - 504x + 63 = 0 \cr & x = \frac{3} {7} \vee x = \frac{3} {{11}} \vee x = \frac{3} {{13}} \cr} $
Opgelost...