Expert: Scott A Wilson - 9/7/2011

Question
Dear Prof Scott

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

As seen from the definition, a quadratic equation is a polynomial equation of the second degree. The general form is

ax^2+bx+c=0,

where x represents a variable or an unknown, and a, b, and c are constants with a ≠ 0. (If a = 0, the equation is a linear equation.)

The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" comes from quadratus, which is the Latin word for "square".


We can compute the real roots of a quadratic equation with the formula for x1 and x2 with D(Discriminant) = Delta = b^2 - 4ac

Question 1
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What are the applications of Quadratic Equations ?. i.e where they can be applied ?

Question 2
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Similar to Quadratic Equations, can we also have say tresatic equation i.e a polynomial equation of the third degree.

The general form would be

ax^3+bx+c=0,

where x represents a variable or an unknown, and a, b, and c are constants with a ≠ 0. (If a = 0, the equation is a linear equation.)

The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. The term "tresatic" is for the Latin word "three". And then we calculate real roots of a tresatic equation , If this is valid what would be formula for Delta ?

Will Delta = b^3 - 4ac ?

What would be the formula for computing real roots of a tresatic equation ?

Question 3
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If Question 2 is valid then can we consider new equations for polynomial equations of then fourth,fifth,sixth degree and so on ..... ?

Then we find the real roots of the polynomial equation of the nth degree ???

Are Question 2 and Question 3 can be considered valid ?


Thanks & Regards,
Prashant S Akerkar

Answer
1) Read http://plus.maths.org/content/101-uses-quadratic-equation to see uses.

2) I have seen the solution of the tresatic equation,
but it was so messy it took a whole period to explain.
I have never seen an application that refers to the Delta term.

3) I also remember in college a senior level course in which we discussed solving polynomials where the highest power was greater than 3.  The only way it could be done back in the early 80's was in specific cases.

Question 2 and Question 3 are valid questions, but the answer to 2 is complicated and the answer to 3 is there is no way to factor them.