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Cubic Equation
Guadalupe Marin-Aguilar
Tania Kristell Contreras
A Step Back in History….
Cubics in Babylonia (2000 BC)
Babylonian calculated cubic roots from tables, but not completely clear of
their methods used.
It is anyhow known that they used a method of root extraction involving only
integer additions.
The table is made out of clay, size, 7,6x4,4x2,3 cm, and contains 3 columns,
30 line in cuneiform script. The only similar text known before is a Late
Babylonian table text.
The Babylonians divided the day into 24hrs, each hour into 60mins,
And each minute into 60secs.
www.groups.dcs.org
Gerolamo Cardano
• His major fields of study were astrology, music, philosophy,
and medicine.
• 131 works were published during his lifetime.
• In mathematics, he wrote a variety of subjects. He wrote a
Book on Games of Chance. The work broke ground on theory
of probability.
Cardano is reported to have correctly predicted to exact
date of his own death.
• His greatest work was Ars Magna (The Great Art) published in
1545. It was the first Latin treatise devoted exclusively to
Algebra and is where the he also published the cubic formula.
Linda Hall Library History of Science Collection
Solving
• You need to solve:
ax³ + bx² + cx + d
Where a ≠ 0
With a=0 the equation is reduced to a
quadratic equation.
• The solution can be integer, real, rational,
irrational or complex number.
How to solve
• You can use the Cardano’s formula which gives
you the roots to the function.
Possible Solutions
• The equation can contain:
- 3 different real solutions
- 2 real solutions, one of them is a
double solution.
- A single real solution which is a triple solution.
- A single real solution and a pair of conjugate
solutions which are complex numbers.
Derivative
• The derivative
will yield
when
.Bearing its resemblance to the quadratic formula,
this formula can be used to find the critical points of a cubic
function. It turns out that, if
, then the cubic function
will have two critical points — a local maximum and a local
minimum; if
, then there is one critical point, and it
will yield the inflection point; if
, then there are no
critical points; the cubic is then strictly monotonic.
• Monotonic  f(x) < f(y)  f(x) preserves the order
 f(x) > f(y)  f(x) reserves the order
The discriminant is positive( not zero).
The discriminant is positive( = zero).
The discriminant is negative.