Solving Cubic Equations

1
Solving Cubic Equations

• Ben Anderson
• Jeff Becker

2
Warm-up Activity

• Use provided ruler and compass to find 1/3 of the
  given angles.

3
Explanation of Activity

• Many early solutions involved
  geometric representations.
• Later answers involved arithmetic without
  concepts of negatives zero.
• Even as math evolved, solutions involving square
  roots of negative numbers eluded mathematicians.

4
Explanation of where the equation 4x3 - 3x a
0 comes from

• Let ? 3a, so cos(?) cos(3a) and let a
  cos(?), then
• a cos(?) cos(3a) cos(2a a)
• cos(2a)cos(a) sin(2a)sin(a)
• cos2(a)-sin2(a)cos(a) 2sin(a)cos(a)sin(a
  )
• cos3(a) sin2(a)cos(a) 2sin2(a)cos(a)
• cos3(a) 3sin2(a)cos(a)
• cos3(a) 31-cos2(a)cos(a)
• cos3(a) 3cos(a) 3cos3(a)
• 4cos3(a) 3cos(a)
• Thus, 4cos3(a) 3cos(a) a 0.
• Finally, setting x cos(a) gives us 4x3 - 3x a
  0.

Trig Property in use here cos(xy)
cos(x)cos(y) - sin(x)sin(y)
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Early Approaches to Solving the Cubic Equation

• Arabic and Islamic mathematicians
• Formulating the Problem
• - Islamic mathematicians, having read major
  Greek texts, noticed certain geometric problems
  led to cubic equations.
• - They solved various cubic equations in the
  10th and 11th centuries using the Greek idea of
  intersecting conics.

6
Arabic and Islamic Mathematicians

• Umar ibn Ibrahim al-Khayyami
• - known in the West as Omar Khayyam
• - systematically classified and solved all types
  of cubic equations through the use of
  intersecting conics
• - publishes Treatise on Demonstrations of
  Problems of Al-jabr and Al-muqabala, devoted to
  solving the cubic equation
• - classifies various forms possessing a positive
  root and 14 other cases not reducible to
  quadratic or linear equations

7
Arabic and Islamic Mathematicians

• Sharaf al-Din al-Tusi
• - born in Tus, Persia
• - He classified cubic equations into several
  groups that differed from the ones Omar
  conceived
• 1. equations that could be reduced to quadratic
  ones, plus x3 d.
• 2. eight cubic equations that always have at
  least one (positive) solution
• 3. types that may or may not have (positive)
  solutions, dependent upon values for the
  coefficients, which include
• x3 d bx2, x3 d cx, x3 bx2 d cx, x3
  cx d bx2, and x3 d b x2 cx
• (his study of this third group was his most
  original contribution)

8
Italian Invention and Dispute

• algebra and arithmetic develop in 13th century in
  Italy through such publications as Leonardo of
  Pisas Liber Abbaci
• little progress is made toward solving the cubic
  equation until the 16th century when the Italian
  theater is set to stage a new production

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Solving the Cubic Equation A Bad Sixteenth
Century Italian Play

• List of Characters
• Scipione del Ferro, the dying scholar
• Antonio Maria Fiore, his student
• Niccolò Fontana (Tartaglia or the Stammerer),
  rival of Fiore
• Girolamo Cardano, new rival to Tartaglia
• Lodovico Ferrari, Cardanos pupil
• Rafael Bombelli, the imaginary fool

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Act I Scene i

• Ferro Fiore, Fiore! Alas, I am dying, Fiore,
  and need to pass on my method for solving the
  cubic equation.
• Fiore (aside) An equation, he speaks? Though I
  am not a great mathematician, I will learn his
  method and someday reveal it to claim great fame.

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Act I Scene ii

• Meanwhile, Tartaglia also knew a method for
  solving the cubic equation
• Tartaglia Though I know how to solve the cubic
  equation, I will not reveal how. This gives me
  the ability to challenge others with a type of
  problem they cant solve. Rich patrons support
  me while I am defeating other scholars in public
  competitions!
• Fiore I have heard your claim of knowing the
  solution to the cubic equation. I challenge you
  to a competition!

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Act I Scene ii continued

• Tartaglia Nave! Willingly I accept!
• Fiore (aside) Victory is within reach, for
  Tartaglia is known as The Stammerer and
  couldnt act in a Bad Sixteenth Century Italian
  Play if he tried!

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Act I Scene ii continued

• The contest ensues
• Tartaglia It appears you only know how to solve
  equations of the form x3 cx d.
• Fiore That is strange, for you only know how to
  solve equations of the form x3 bx2 d.
• Tartaglia Well, no matter. I have prepared
  fifty more unrelated homework problems for you.
• Fiore Argh!!

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Act I Scene ii continued

• Tartaglia wins the competition
• Fiore It appears my knowledge of mathematics
  does not extend beyond cubic equations, yet you
  have managed to find solutions to all my
  problems.
• Tartaglia I forgive your ignorance. I will
  decline the prize for my victory, which included
  thirty banquets hosted by you, loser, for me, the
  winner, and all my friends.
• Fiore recedes into the obscurity of history.

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Act II Scene i

• Cardano Greetings. I am a great doctor,
  philosopher, astrologer, and mathematician.
  Please give me the solution to the cubic
  equation.
• Tartaglia Okay, but you must swear never to
  reveal my secret.
• Cardano I swear.

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Act II Scene i continued

• Within the next decade, Cardano publishes the Ars
  Magna, containing complete solutions to solving
  any cubic equation. Included are geometric
  justifications for why his methods work. He
  includes a subtle footnote to Tartaglia that del
  Ferro had discovered the crucial solution before
  him, justifying his publication of del Ferros
  work. Cardano also includes a solution for the
  quartic, which his pupil Ferrari devised.

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Act II Scene ii

• While Tartaglia is furious, Ferrari contacts him
  to challenge him to a competition. Tartaglia
  refuses until he is offered a professorship in
  1548 on the condition that he defeats Ferrari in
  the contest.
• Ferrari I know how to solve the general cubic
  and quartic equations. Tartaglia may not have
  read Cardanos book on those equations, which
  contains a solutions manual.

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Act II Scene ii continued

• Tartaglia I dont like books, especially when
  its other peoples bad writing, but I like math
  competitions and I assume Ill win easily. This
  time I think I will accept my victory spoils!
• Tartaglia loses and remains resentful of Cardano
  for the rest of his life

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Act II Scene ii Continued

• This is not the end of the story.
• Some expressions that resulted from Cardanos
  method in equations of the form x3 px q
  didnt make sense.
• For x3 15x 4, Cardanos method produces

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Act II Scene iii

• Bombelli For the expression x3 px q, there
  is always a positive solution, regardless of the
  positive values of p and q. For x3 15x 4,
  this would be x 4. However, for many values of
  p and q, solving the equation gives square roots
  of negative numbers. Hence, I will legitimize
  these numbers by calling them imaginary
  numbers, making myself nothing short of a genius
  in my time!!!

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Play Conclusion

• The next target, equations of degree 5, proves
  more difficult, turning a different chapter in
  history when abstract algebra rears its head.

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Procedure for Solving Cubic Equations

• Beginning with an equation of the form
• ax3 bx2 cx d 0
• Substitute x y b/3a
• a(y b/3a)3 b(y b/3a)2 c(y b/3a) d
  0
• and simplifying gives
• ay3 b2y/3a cy 2b3/27a2 bc/3a d 0
• Make equation into the form y3 Ay B
• y3 (c/a - b2/3a2)y (bc/3a2 d/a - 2b3/27a3)

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Procedure Continued

• Find s and t such that 3st A (Equation 1)
• and (s3 - t3) B (Equation 2)
• Fact y s t is a solution to the cubic of
  the
• form y3 Ay B
• To find s and t, we solve Equation 1 in terms of
  s and substitute into Equation 2.
• (A/3t)3 t3 B
• Through algebra, we obtain
• (A3/27t3) t3 B
• t6 Bt3 A3/27 0
• Substitute u t3
• u2 Bu A3/27 0
• Quadratic formula gives us value of u.

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Procedure Concluded

• We then use this value of u to obtain t, which
  in turn is used to find s. Next, we use the fact
  (from the previous page) that y s t is a
  solution to the cubic and plug s t into the
  original substitution of x y b/3a to find the
  first real root. To find the other roots (real or
  imaginary) of the equations, we use this solution
  to reduce the cubic equation into a quadratic
  equation by long division. At this point, we can
  use the quadratic formula to obtain the other
  roots of the equation.

25
Present Day

• Today there exists numerous cubic equation
  calculators that solve cubics at the click of a
  mouse. One such example is www.1728.com/cubic.htm
  

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Timeline

• 400 B.C. - Greek mathematicians begin looking at
  cubic equations
• 1070 A.D. - Al-Khayammi publishes his best work
  Treatise on Demonstrations of Problems of Aljabr
  and Al-muqabala
• Late 12th century Sharaf continues
  Al-Khayammis work and adds new solutions
• 14th century - Algebra reaches Italy
• Early 16th century del Ferro and Tartaglia
  discover how to solve certain cubics but keep
  their solutions secret

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Timeline continued

• 1535 Fiore challenges Tartaglia to a
  competition involving cubic equations, and
  Tartaglia wins. News of his victory reaches
  Cardano.
• 1539 - Tartaglia explains his partial solution to
  Cardano.

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Timeline continued

• 1545 Cardano produces a complete solution to
  cubic equations and publishes it in Ars Magna,
  which also includes Ferraris solution to the
  quartic.
• Late 16th century Bombelli introduced the idea
  of using imaginary numbers in the solution to
  cubic equations

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References

• Berlinghoff and Gouvea. Math Through the Ages.
• Katz, Victor J. A History of Mathematics.
• Cubic Equation Calculator. www.1728.com/cubic.htm
  
• Cubic Equations. en.wikipedia.org/wiki/Cubic_equa
  tion
• The Cubic Formula. http//www.sosmath.com/algeb
  ra/factor/fac11 /fac11.html