Quadratic, Cubic, and Quartic Equations

1. Quadratic, Cubic, and Quartic Equations By Robert Quin (raq5005), Brian Thorwarth (bjt5082)

2. 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.

3. 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 (17th 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.

4. 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.

5. 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.

6. brahmagupta • The most well known Indian mathematician of the seventh century, Brahmagupta, developed Babylonian methods into an almost modern method • He used abbreviations for the unknown, generally based on the color that was used. Here is a quote by Brahmagupta describing how to solve a quadratic and a famous quadratic equation equation. “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, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.” (Brahmasphutasiddhanta (Colebrook translation, 1817, page 346

7. Abu Ja'far Muhammad ibn Musa Al-Khwarizmi • Al'Khwarizmi was an Islamic mathematician who developed six chapters of quadratics (though his contains no zeroes or negatives) • Squares equal to roots • Squares equal to numbers • Roots equal to numbers. • Squares and roots equal to numbers • Squares and numbers equal to roots • Roots and numbers equal to squares

8. Abraham bar HiyyaHa-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 equationx2 - ax + b = 0

9. Descartes and Leibniz • Rene Descartes- in 1637- published La Giometrie giving us the quadratic formula in the form we know today. • Leibniz (in the picture)-in 1673- Developed the first true algebraic proof, for cubic equations. All other proofs were geometric in nature.

10. 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.

11. 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.