Expert: Paul Klarreich - 9/7/2011

Question
QUESTION: Dear Prof Paul

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
------------

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

Question 2
------------

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
------------

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: Questioner:Prashant S Akerkar
Country:Maharashtra, India
Category:Calculus
Private:No
Subject:Quadratic Equation
Question:

Dear Prof Paul

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
------------

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

>> For this, you have to consult your teacher.  There are many.

Question 2
------------

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

>>>>>>>>>>>> It's called a CUBIC.

The general form would be

ax^3+bx+c=0,

>>>>>>>>> ax^3 + bx^2 + cx + d = 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 ?

>>>> Look up Ferrari's formula. (or is it Cardano's?  I forget.)

See:

mathworld.wolfram.com/CubicFormula.html

Question 3
------------

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 ?

>>>>>>>>>>  Look up Galois. You will find there is a QUARTIC formula.  But there is no

formula for n > 4.  [I didn't say it hasn't been found; I said there is no formula.  Don't waste your time trying to find it.]

Thanks & Regards,
Prashant S Akerkar


---------- FOLLOW-UP ----------

QUESTION: Dear Prof Paul

Thank you.

Also I found a Link on Quintic Equation -> Polynomial of Five Degree.

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


What it suggests is we can find roots of a

Quadratic Equation - Polynomial of Second Degree,
Cubic Equation - Polynomial of Three Degree
Quintic Equation - Polynomial of Five Degree

so we can find roots of Polynomials with n=1,2,3,4,5,6 ..........

More the Degree in a Polynomial, more could be the complexity in finding out the roots.

isn't it ?

i.e Is it possible to find Roots of a Polynomial where n=100 i.e Degree is 100 i.e x1,x2,x3 ....... x100 ?

Can we write the Law for Computing Roots of a Polynomial of nth Degree ?

The Complexity in finding out the roots of a Polynomial of nth Degree is directly proportional to Increasing value of Degree of a Polynomial.

Any Amendments to be made to the above law ?

Thanks & Regards,
Prashant S Akerkar

Answer
Note: I do not accept private questions.  I change them to public, as I note in my instructions.

QUESTION: Dear Prof Paul

Thank you.

Also I found a Link on Quintic Equation -> Polynomial of Five Degree.

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


What it suggests is we can find roots of a

Quadratic Equation - Polynomial of Second Degree,
Cubic Equation - Polynomial of Three Degree
Quintic Equation - Polynomial of Five Degree

>>>>>>>>>>>> Read it again.  Note that it says "we can find roots of SOME quintic equations."  And also that in general,

we can find roots of Polynomials with n = 5,6 only if they meet certain conditions, such as being factorable.  

And note the Abel-Ruffini theorem, which is the proof that a general equation of degree 5 or higher is not solvable in radicals.  ('Solvable in radicals' means there is a general formula for the roots, such as the quadratic formula, giving an exact expression for the roots.)

I suggest that when you get past your three semesters of calculus, you take a year of Abstract Algebra (a.k.a. Modern Algebra, a.k.a. Groups, Rings, and Fields, BUT NOT A.K.A. College Algebra.)  There you will see the exciting proof of this theorem, and a proof of the Fundamental Theorem of Algebra.)
.......................................

More the Degree in a Polynomial, more could be the complexity in finding out the roots.

isn't it ?

i.e Is it possible to find Roots of a Polynomial where n=100 i.e Degree is 100 i.e x1,x2,x3 ....... x100 ?

Can we write the Law for Computing Roots of a Polynomial of nth Degree ?

The Complexity in finding out the roots of a Polynomial of nth Degree is directly proportional to Increasing value of Degree of a Polynomial.

Any Amendments to be made to the above law ?