2011-2012 Mathematics 2nd Wordpress Project
Name
: Musa
Surname
: Çelik
Class
: 9-A
No
: 228
Topic: Quadratic Equations and
Polynomials
Quadratic Equations and Polynomials
Life is based many different things, but science is the most basic thing of the life.
Because people wonder about what is happening around them, and they are looking for
answers to the questions that begin with why. Actually, they are looking for logics, reasons.
Mathematics provides them to find their answers. Mathematics is the life and science itself.
One of the most important topics of the mathematics is functions. We use functions in
daily life. Actually we use multi-functions. Those multi-functions get bigger and more
complex, so then we call them polynomials with degrees. But we use a simple type of
polynomials, which consist of quadratic equations. Those equations are with degree two as
you understand from the word “quadratic”. Quadratic equations and polynomials are very
close topics. Every quadratic equation is a polynomial, but not every polynomial is quadratic.
As we deal with x and ys people from very former times wondered how to solve these kinds
of equations. For this reason people reinforced many techniques to solve those types of
polynomials. One way of solving functions (polynomials) with degree 2 is quadratic formula,
for example. And many people got themselves crazy while trying to find the ways for
solutions.
As you could understand, many people form BC to now on work on quadratic
equations and polynomials. As a result it was so hard to deliver those ways to here. I can give
many examples to the works but I can give you some works and the scientists who worked on
these topics and who developed for us:
Quadratic equations
Even though the Babylonians didn't have any notion of what an 'equation' is, they found the first
algorithmic approaches to problems, which would give rise to a quadratic equation today. Their
method is essentially one of completing the square. However all Babylonian problems had answers
which were positive (more accurately unsigned) quantities since the usual answer was a length.
The first known solution of a quadratic equation is the one given in the Berlin papyrus from the
Middle Kingdom (ca. 2160-1700 BC) in Egypt.
This problem reduces to solving
x2 + y2 = 100
y = 3/4 x
In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used
it to solve quadratic equations, amounted to finding a length which in our notation was the root of a
quadratic equation. Euclid had no notion of equation, coefficients etc. but worked with purely
geometrical quantities. His Data contains three problems involving quadratics. In his work
Arithmetica, the Greek mathematician Diophantus (ca. 210-290) solved the quadratic equation, but
giving only one root, even when both roots were positive.Hindu mathematicians took the Babylonian
methods further. Aryabhata (475 or 476-550) gave a rule for the sum of a geometric series that
shows knowledge of the quadratic equations with both solutions.Later, Brahmagupta (598-665 AD)
gives an, almost modern, method which admits negative quantities. He also used abbreviations for
the unknown, usually the initial letter of a colour was used, and sometimes several different
unknowns occur in a single problem. The Arabs did not know about the advances of the Hindus so
they had neither negative quantities nor abbreviations for their unknowns. However al-Khwarizmi (c
800) gave a classification of different types of quadratics (although only numerical examples of each).
The different types arise since al-Khwarizmi had no zero or negatives. He has six chapters each
devoted to a different type of equation, the equations being made up of three types of quantities
namely: roots, squares of roots and numbers i.e. x, x2 and numbers.
Al-Khwarizmi solves each type of equation:
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
- essentially the familiar quadratic formula given for a numerical example in each case, and then a
proof for each example which is a geometrical completing the square.
Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book Liber
embadorum published in 1145 which is the first book published in Europe to give the complete
solution of the quadratic equation.
ViËte was among the first to replace geometric methods of solution with analytic ones, although
he apparently did not grasp the idea of a general
quadratic equation
Part Two: Solving The Quadratic (Theory)
A quadratic equation is a second-order polynomial equation in a single variable x
(with a≠0).
Because it is a second-order polynomial equation,
the fundamental theorem of algebra guarantees
that it has two solutions. These solutions may be
both real, or both complex.
The roots x can be found by completing the square,
Polynomials
ca. 2000 BC
Babylonians solve quadratics in radicals.
ca. 300 BC
Euclid demonstrates a geometrical construction for solving a quadratic.
ca. 1000
Arab mathematicians reduce:
ux2p + vxp = w
to a quadratic.
1079
Omar Khayyam (1050-1123) solves cubics geometrically by intersecting parabolas and circles.
ca. 1400
Al-Kashi solves special cubic equations by iteration.
1484
Nicholas Chuqet (1445?-1500?) invents a method for solving polynomials iteratively.
1515
Scipione del Ferro (1465-1526) solves the cubic:
x3 + mx = n
but does not publish his solution.
1535
Niccolo Fontana (Tartaglia) (1500?-1557) wins a mathematical contest by solving many different
cubics, and gives his method to Cardan.
1539
Girolamo Cardan (1501-1576) gives the complete solution of cubics in his book, The Great Art, or the
Rules of Algebra. Complex numbers had been rejected for quadratics as absurd, but now they are
needed in Cardan's formula to express real solutions. The Great Art also includes the solution of the
quartic equation by Ludovico Ferrari (1522-1565), but it is played down because it was believed to be
absurd to take a quantity to the fourth power, given that there are only three dimensions.
1544
Michael Stifel (1487?-1567) condenses the previous eight formulas for the roots of a quadratic into
one.
1593
Francois Viete (1540-1603) solves the casus irreducibilis of the cubic using trigonometric functions.
1594
Viete solves a particular 45th degree polynomial equation by decomposing it into cubics and a
quintic. Later he gives a solution of the general cubic that needs the extraction of only a single cube
root.
1629
Albert Girard (1595-1632) conjectures that the nth degree equation has n roots counting multiplicity.
1637
Rene Descartes (1596-1650) gives his rule of signs to determine the number of positive roots of a
given polynomial.
1666
Isaac Newton (1642-1727) finds a recursive way of expressing the sum of the roots to a given power
in terms of the coefficients.
1669
Newton introduces his iterative method for the numerical approximation of roots.
1676
Newton invents Newton's parallelogram to approximate all the possible values of y in terms of x, if:
Sigma(i, j = 0 -> n) [aij xiyj] = 0
1683
Ehrenfried Walther von Tschirnhaus (1646-1716) generalizes the linear substitution that eliminates
the xn-1 term in the nth degree polynomial to eliminate the xn-2 and xn-3 terms as well. Gottfried
Wilhelm Leibniz (1646-1716) had pointed out that trying to get rid of the xn-4 term usually leads to a
harder equation than the original one.
1691
Michael Rolle (1652-1719) proves that f'(x) has an odd number of roots in the interval between two
successive roots of f(x).
1694
Edmund Halley (1656-1742) discusses interative solutions of quartics with symbolic coefficients.
1728
Daniel Bernoulli (1700-1782) expresses the largest root of a polynomial as the limit of the ratio of the
successive power sums of the roots.
1732
Leonard Euler (1707-1783) tries to find solutions of polynomial equations of degree n as sums of nth
roots, but fails.
1733
Halley solves the quadratic in trigonometric functions.
1748
Colin Maclaurin (1698-1746) generalizes Newton's relations for powers greater than the degree of
the polynomial.
1757
Johann Heinrich Lambert (1728-1777) gives series solutions of trinomial equations:
xp + x + r = 0
1762
Etienne Bezout (1730-1783) tries to find solutions of polynomial equations of degree n as linear
combinations of powers of an nth root of unity, but fails.
1762
Euler tries to find solutions of polynomial equations of degree n as linear combinations of powers of
an nth root, but fails.
1767
Joseph Louis Lagrange (1736-1813) expresses the real roots of a polynomial equation in terms of a
continued fraction.
1769
Lagrange expands a function as a series in powers of another function and uses this to solve trinomial
equations.
1770
Lagrange shows that polynomials of degree five or more cannot be solved by the methods used for
quadratics, cubics, and quartics. He introduces the Lagrange resolvent, an equation of degree n!.
1770
Euler gives series solutions of:
xm+n + axm + bxn = 0
1770
John Rowning (1699-1771) develops the first mechanical device for solving polynomial equations.
Although the machine works for any degree in theory, it was only practical for quadratics.
1771
Gianfrancesco Malfatti (1731-1807), starting with a quintic, finds a sextic that factors if the quintic is
solvable in radicals.
1772
Lagrange finds a stationary solution of the three body problem that requires the solution of a quintic.
1786
Erland Samuel Bring (1736-1798) proves that every quintic can be transformed to:
z5 + az + b = 0
1796
Jean Baptiste Joseph Fourier (1768-1830) determines the maximum number of roots in an interval.
1799
Paolo Ruffini (1765-1822) publishes the book, General Theory of Equations, in which the Algebraic
Solution of General Equations of a Degree Higher than the Fourth is Shown to Be Impossible.
1799
Carl Friedrich Gauss (1777-1855) proves the fundamental theorem of algebra: Every nonconstant
polynomial equation has at least one root.
1801
Gauss solves the cyclotomic equation:
z17 = 1
in square roots.
1819
William George Horner (1768-1847) presents his rule for the efficient numerical evaluation of a
polynomial. Ruffini had proposed a similar idea.
1826
WilNiels Henrik Abel (1802-1829) publishes Proof of the Impossibility of Generally Solving Algebraic
Equations of a Degree Higher than the Fourth.
1871
Ludwig Sylow (1832-1918) puts the finishing touches on Galois's proofs on solvability.
1873
Hermann Amandus Schwarz (1843-1921) investigates the relationship between hypergeometric
differential equations and the group structure of the Platonic solids, an important part of Klein's
solution to the quintic.
1877
Felix Klein (1849-1925) solves the icosahedral equation in terms of hypergeometric functions. This
allows him to give a closed-form solution of a principal quintic.
1884, 1892
Ferdinand von Lindemann (1852-1939) expresses the roots of an arbitrary polynomial in terms of
theta functions.
1885
John Stuart Cadenhead Glashan (1844-1932), George Paxton Young (1819-1889), and Carl Runge
(1856-1927), show that all irreducible solvable quintics with the quadratic, cubic, and quartic terms
missing have a spezial form.
1890, 1891
Vincenzo Mollame (1848-1912) and Ludwig Otto Hoelder (1859-1937) prove the impossibility of
avoiding intermediate complex numbers in expressing the three roots of a cubic when they are all
real.
1891
Karl Weierstrass (1815-1897) presents an interation scheme that simultaneously determines all the
roots of a polynomial.
1892
David Hilbert (1862-1943) proves that for every n there exists an nth polynomial with rational
coefficients whose Galois group is the symmetric group Sn
1937
Scientists at Bell Labs build the Isograph, a precision instrument that calculates roots of polynomials
up to degree 15.
1938, 1942
Emil Artin (1898-1962) uses field theory to develop the modern theory of algebraic equations.
1957
Vladimir Arnol'd, using results of Andrei Kolmogorov (1903-1987), shows that it is possible to express
the roots of the reduced 7th degree polynomial in continuous functions of two variables, answering
Hilbert's 13th problem in the negative.
1984
Hiroshi Umemura expresses the roots of an arbitrary polynomial through elliptic Siegel functions.
1989
Peter Doyle and Curt McMullen construct a generally convergent, purely iterative algorithm for the
numerical solution of a reduced quintic, relying on the icosahedral equation.
1991, 1992
David Dummit and (independently) Sigeru Kobayashi and Hiroshi Nakagawa give methods for finding
the roots of a general solvable quintic in radicals.