Quadratic, cubic, and quartic equations

By: David Panson

Quadratic Equations

• Equations with an X2 set to 0

• General formula for the equation is

ax2+bx+c=0

• General Equation for solution is

Cubic Equations • Equations with an X3 set to 0

• General formula for the equation is

ax3+bx2+cx+d=0

Quartic Equations

• Equations with an X4 set to 0

• General formula for the equation is

ax4+bx3+cx2+dx+e=0

The Challenge • Solve with basic operations that cancel and

prevent the use of mind blowing math in a pre calculator era

• Be able to account for negatives and zeros

• Make a uniform way to solve all variations of each problem

Solutions

• Quartic and Cubic

– Roots!

• Quadratic

– Factoring

– Quadratic formula

– Completing the square

Use Today • Math class? And….yeah.

• Quadratic, cubic, and quartic problems are solved still so they most matter right?

• They are needed to understand acceleration of cars, planes, football players, or whatever else, air flow and flight, cell phone signal technology, and tons of other stuff.

• Like math in general they can be found most anywhere. Even hiding in science.

Timeline

• 400 BC Sumerians Start algorithmic thoughts • 300 BC Euclid and his geometry stuff • 600 AD Brahmagupta finds out he can have less than 0 friends (but only

in India.)

• 800 Al Khwarizmi and his 6 class acts • 1145 Savasorda and his best friend. A book. • 1494 Luca Pocioli spreads Suma knowledge • 1515ish Scipione dal Ferro looks at a half full cup. But he thinks its

full.

• 1526 Ferro finally shares his secret… with a stupid genius (Fior)

• 1545 Cardan breaks the bro code. Politely. But Tartaglia is still sad. • 1567 Stifel does a magic trick. Nerd style. • 1637 Descartes makes it FBO… or you know. at least O. • 1673 Leibniz proves that centuries of work is in fact not useless.

Al Khwarizmi’s Classifications of Quadratics

• Squares equal to roots.

• Squares equal to numbers.

• Roots equal to numbers.

• Squares and roots equal to numbers, e.g. x2 + 10x = 39.

• Squares and numbers equal to roots, e.g. x2 + 21 = 10x.

• Roots and numbers equal to squares, e.g. 3x + 4 = x2.

Fior VS Tartaglia

• 30 Problems to be solved in 50 days.

• Fior issued the challenge

• Tartaglia solves all problems super fast(like a boss)

– They were all x3 + mx2 = n form

Timeline for People that aren’t me • 400 BC Sumerians Start algorithmic problem solving • 300 BC Euclid used geometry to answer some quadratics • 600 AD Brahmagupta finds near modern methods but only in India • 800 Al Khwarizmi and indentifies 6 classifications of quadratics. • 1145 Savasorda published Liber Embadorum first published

quadratic method • 1494 Luca Pocioli published Suma unified previous knowledge • 1515ish Scipione dal Ferro solves cubics but only some. • 1526 Ferro finally shares his secret solution with Fior

• 1545 Cardan published Ars Magna and Tartaglias cubic equation • 1567 Stifel condensed previous knowledge into the quadratic

equation • 1637 Descartes published La Giometrie with the modern formula. • 1673 Leibniz proves the equation with a purely algebraic proof.

Works by smarter people • http://www-history.mcs.st-

andrews.ac.uk/HistTopics/Quadratic_etc_equations.html

• http://plus.maths.org/content/101-uses-quadratic-equation-part-ii

• http://dwick.org/pages/cubicquartic.pdf

• http://archives.math.utk.edu/ICTCM/VOL16/C037/paper.pdf

• http://en.wikipedia.org/wiki/Quadratic_equation