Mathematics - Quadratic Equation (Part 1)

Picture is obtained from http://foldspace.files.wordpress.com/2008/06/c4743r.jpg

The mathematics question is,
Find the value of k of the quadratic equation

Such that one of its roots is thrice the value of the other.


Solution:
Let α and β be the roots. We have,

Sum of roots:


Product of roots:


Given that,



Substituting into eq. (1), we get



Therefore the other root is,


What to do next?
The roots, α and β are found but in terms of k.
How to find the value of k?
Note that equation (2) is not yet used.


Picture is obtained from http://plus.maths.org/issue29/features/quadratic/Quadratic.jpg

Substituting α and β into eq. (2), we have,




How to obtain the Quadratic Formula

From Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Quadratic_equation)
In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is



The quadratic coefficients are the following:
a is the coefficient of x2
b is the coefficient of x
c is the constant coefficient (also called the free term or constant term).

Quadratic equations are called quadratic because quadratus is Latin for "square".
The roots x or the quadratic formula can be derived by the method of completing the square. Dividing the quadratic equation



by a (which is allowed because a is non-zero), gives:



or



The quadratic equation is now in a form to which the method of completing the square can be applied.
To "complete the square" add the constant,

, to both sides,



The left side is now a perfect square because



Substituting the left side with the perfect square we have,



The right side can be written as a single fraction, with common denominator 4a2. This gives



Taking the square root of both sides yields



Isolating x, gives



This equation is known as the quadratic formula.




How to obtain Sum of roots and Product of roots

The general form is of a quadratic equation is


Dividing the quadratic equation by coefficient, a, we have


If a quadratic equation has roots, α and β, the equation can be written as


That is,



Comparing eq.(1) and eq.(2),




Reference

• http://en.wikipedia.org/wiki/Quadratic_equation
• Additional Mathematics, 8th Edition, by The Keng Seng, Loh Cheng Yee, Consultant: Dr. Yeap Ban Har, Shinglee Publishers Pte Ltd, Chapter 2 - “Quadratic Equations and Functions”, Page 32
• http://foldspace.files.wordpress.com/2008/06/c4743r.jpg (Blackboard)
• http://plus.maths.org/issue29/features/quadratic/Quadratic.jpg (Scary)