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ADAM CLEMENTS SME 430 Discussion Post #9 3/31/2010

Complete the readings for this week and then post your reflections to the prompts below:

Quadratic Equations

What is meant by quadratic equations?o A quadratic equation is when at least one of the variables in an equation is

unknown and is squared.

What are differences between todays approaches to solving quadratic equationsand the approaches used in the past?

o In todays approach to solving the quadratic equation, we consider both thepositive and negative square roots.

o Also in todays approach, we call the unknown x, and call the square x2. Thus weuse letters to denote the equation where in the past it was spelled out in words.

Create your own quadratic equation.o x2 +10x=56

Describe how you would solve this quadratic equation using a geometric method,similar to Al-Khwarizmis approach.

o Draw a square with side lengths x by x giving us an area of x2o Draw an attached share with side lengths x by 10 giving us an area of 10xo Divide the 10x rectangle in half and move this half to the other bottom side of the

x2.

o Now compute the area of the left over square (5 x 5 = 25)o

Now you have a large square of lengths x +5 by x+5 giving you areas of x

2

+2(5x) +25

o The total area is still 56 and we add the 25 so 56+25=81o This means that its side is equal to the square root of 81 which is 9.o So since it is the side of the bi square is x + 5, we conclude that x + 5 = 9 so x = 4.

What are the advantages of setting all quadratic equations equal to zero?o ax2 + bx = c and ax2 + c = bx could be seen as special cases of the general

equation ax2

+ bx + c = 0

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Solving Cubic Equations

Describe the culture of mathematics in Europe in the 15th and 16th centuries.o Much of the culture of mathematics was very secretive. Mathematicians would

discover how to solve a problem, but then keep their solutions secret. Because of

this competition system, it encourage people to keep secrets.

The chapter notes that there are 14 different ways of describing cubic equations (p.110). Write 5 of these ways.

o x3 + ax = bo x3 = ax + bo x3 b = axo x3 - ax = bo x3 + ax b = 0

Describe how the solution of the cubic equation eventually contributed to work onimaginary numbers.

o When Cardanos method was applied to the equation of the from x3 = px + q,often you got an answer that didnt make sense. Later Bombelli began discussingthe equation and showed geometrically that it always had a positive answer. But

on the other hand, se showed that solving this equation led to square roots ofnegative numbers. He showed that it is possible to work with square roots of

negative numbers and still get reasonable answers, thus leading to work on

imaginary numbers.

The book notes that the discovery of the solution to the cubic equation led toadditional discoveries regarding quartic equations and quintic equations. Whatwere the discoveries that were made about these two new types of equations?

o Lodoviso Ferrari applied ideas to the general equation of degree four (the quartic)and managed to find a solution for that too.

o With the cubic and quartic solved, the natural next target was the equation ofdegree 5. That proved to be a much more difficult problem. In fact, it turned out

to be impossible to find a formula for solving the general quintic equation.

Proving this required a complete change of point of view, which eventually led tothe development of abstract algebra.