Transcript Quadratic, Cubic, and Quartic Equations

DEVELOPMENT OF QUADRATIC EQUATIONS
The development of quadratic equations involved many contributions from many
different people. Starting with the extremely basic Babylonian, “equation-less”,
completing the square technique, to the discovery of negative and imaginary
numbers, to Leibniz’s contributions in the 17th century regarding cubic equations.
The improvements and eventual understanding of quadratic and cubic equations
was an evolutionary process that took centuries to develop and use in everyday
life.
MATH BEFORE QUADRATIC EQUATIONS
Before people understood the power of quadratic equations, society was extremely
limited in its mathematic abilities. The quadratic formula is extremely important in
calculus, physics, algebra, and all other math based subjects. Simple ideas like
finding the lengths of the sides of triangles algebraically was impossible until
quadratic equations came along.
The process of understanding the use and power of quadratic equations took
centuries to complete beginning with the idea of, completing the square, with the
Babylonians (400 B.B.) to Leibniz’s contribution to cubic equations (17 th century).
Ultimately, the use and understanding of quadratic equations was a necessary
process in history to further develop our knowledge of algebra, calculus, and our
world as we know it today.
ORIGIN
Babylonians were believed to be the
first to solve quadratic equations
around 400 BC
However, they had no idea of what an
equation was, but they did know
an approach that would later be
used to solve quadratics,
completing the square.
The Babylonians created these
equations although they didn’t
understand the idea of an
equation.
EUCLID
Around 300 BC the Greek
Mathematician Euclid developed a
geometrical method that later
mathematicians would use to solve
quadratic equations
His method led to finding what was in our
notation was the root of a quadratic
equation
Euclid developed a strictly geometrical
way to solve a quadratic equation.
BRAHMAGUPTA
The most well known Indian
mathematician of the seventh
century, Brahmagupta, developed
Babylonian methods into an almost
modern method
“To the absolute number multiplied by
four times the [coefficient of the] square,
add the square of the [coefficient of the]
middle term; the square root of the same,
He used abbreviations for the
less the [coefficient of the] middle term,
unknown, generally based on the
color that was used. Here is a quote being divided by twice the [coefficient of
by Brahmagupta describing how to the] square is the value.”
(Brahmasphutasiddhanta (Colebrook
solve a quadratic and a famous
quadratic equation equation.
translation, 1817, page 346
ABU JA'FAR MUHAMMAD IBN MUSA ALKHWARIZMI
Al'Khwarizmi was an Islamic mathematician
who developed six chapters of
quadratics (though his contains no
zeroes or negatives)
1.Squares equal to roots
2.Squares equal to numbers
3.Roots equal to numbers.
4.Squares and roots equal to numbers
5.Squares and numbers equal to roots
6.Roots and numbers equal to
squares
ABRAHAM BAR HIYYA HA-NASI
Published Liber embadorum in 1145,
which was the first European book to
give the first solution of the quadratic
equation
It contains the first complete
solution of the quadratic equation
x2 - ax + b = 0
DESCARTES AND LEIBNIZ
Rene Descartes- in 1637published La Giometrie giving
us the quadratic formula in
the form we know today.
Leibniz (in the picture)-in 1673Developed the first true
algebraic proof, for cubic
equations. All other proofs
were geometric in nature.
REAL WORLD APPLICATIONS
Quadratic Equations, the basis of algebra, can be used in an
incredible number of ways.
-
Finding the curve on a cartesian grid, the flight of a ball, the
military uses it to predict where artillery shells will land,
explaining how planets in our solar system revolve around the
sun, Newton’s Laws of motion can be proven using quadratic
equations, police officers use quadratic equations to
determine the velocities of cars during an accident, and
quadratic equations are used in the creation of sound
systems in homes, movie theaters, and arenas.
-
These examples and much more can be done with quadratic
equations.
QUADRATIC EQUATIONS AT PENN STATE
MATH 021: College Algebra I (3:3:0). Quadratic equations; equations in quadratic
form; word problems, graphing; algebraic fractions; negative and rational
exponents; radicals.
College Algebra II (3:3:0). Relations, functions, graphs; polynomial, rational functions,
graphs; word problems; nonlinear inequalities; inverse functions; exponential,
logarithmic functions.
MATH 140 (GQ) CALCULUS WITH ANALYTIC GEOMETRY I (4 semester hours)
Functions, limits; analytic geometry; derivatives, differentials, applications;
integrals applications.
MATH 141 (GQ) Calculus with Analytic Geometry II (4) Derivatives, integrals,
applications; sequences and series; analytic geometry; polar coordinates.
In these courses and many others, including physics courses, understanding of how
to use quadratic equations is a must.