## Sunday, April 19, 2009

### 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.

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

How to obtain the Quadratic Formula

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).

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