Cardans Creature

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Cardans Creature
Ludovico Ferrari
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The Challenge of the Quartic

• Cardans Servant and Pupil
• Taking up for his master
• Contest with Tartaglia

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The Famous Quartic Solution
Step 1 Reduce the Equation, x4 ax3
bx2 cx d 0 by Substituting y a/4 in
for x.
This gives us y4 py2 qy r 0 .
Step 2 Set equation to a workable form
y4 py2 -qy - r
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Problem - Continued
Step 3 Complete the square on the right side. We
do this by adding (py2 p2) to both
sides. This gives Y4 py2 py2 p2
or y4 2py2 p2
which equals (y2 p)2. Therefore we have,
(y2 p)2 py2 p2 qy - r
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Problem continued
Step 4 Ferrari wanted the left side of the
equation into the form (y2 p z) 2 so he
introduces the new variable z by adding 2z(y2
p) z2 to each side. Therefore we have (y2
p z)2 py2 p2 qr r 2z(y2 p)
z2. (p 2z)y2 qy
(p2 r 2pz z2)
Step 5 Now we must make the right hand side a
perfect square which happens when the
discriminate is 0, that is 4(p2z)(p2 r 2pz
z2) q2.
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Problem Continued
Step 6 This requires us to solve a cubic in z,
namely 8z3 20pz2 (16p2 8r)z 4p3
4pr q2) 0. This equation is known as the
resolvent cubic, and it can be solved in the
usual way that has been previously discussed.
The best way to understand the procedure is to
work through an example. (See handout sheet
provided).
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References

• Textbook
• http//www-gap.dcs.st-and.ac.uk/history/Mathemat
  icians/Ferrari.html
• http//www-gap.dcs.st-and.ac.uk/history/Mathemat
  icians/Tartaglia.html
• http//www.karlscalculus.org/quartic.html