What is Sign Scheme? How is it useful?

							
Dear Rohan,
The expression  ax² + bx + c  is called a quadratic expression or a quadratic polynomial or a polynomial of degree two, where a, b, and c have  same meaning as discussed in earlier posts of quadratic equation. Let this expression be denoted by y.

   y = ax² + bx +c

⇒y= a ( x² + (b/a)x + c/a ) 

      = a( x² + 2*(b/2a)*x + (b/2a)² - (b/2a)² + c/a )

      = a{ (x + b/2a)² - ( b² -4ac)/4a² }

   b² -4ac  is denoted by D. Thus,

    y=a{ (x + b/2a)² - D/4a² }         - eq 1

Now consider the following cases: 

case 1: a > 0, and D < 0 

The quantity (x + b/2a)² in eq 1 is always greater than or equal to 0 since it is the square of a real number. As D <0, so (-D) will be >0. And as a>0, 4a² will be greater than zero.  Therefore- 

     sign of y=sign of  a{ +ve + (+ve/+ve) }

or, sign ofy= sign of  a( +ve) 

or, sign ofy= sign of  a 

conclusion: y is always +ve. 

Example:  y = x² + x + 1
                  Here a=1 >0
                            D= 1²- 4*1*1 = -3 <0
So the value of y must be greater than zero for any real value of x, as can be seen below in the graph of the quadratic polynomial  x² + x + 1 

    



case 2: a<0, D<0

The quantity (x + b/2a)² in eq 1 is always greater than or equal to 0 since it is the square of a real number. As D <0, so (-D) will be >0. 4a² is the square of (-2a) and hence will always be > 0, as a <0.( Note: what i am emphasizing by saying a<0 is that ''a'' is not equal to 0. Had a been equal to zero, ''a'' would not have been a quadratic polynomial in the first place)

   sign of y=sign of  a{ +ve + (+ve/+ve) }

or, sign ofy= sign of  a( +ve) 

or, sign ofy= sign of  a 

conclusion: y is always -ve.
Example:  y = -x² + x -1
                  Here a=-1 <0
                            D=  -3 <0
So the value of y must be less than zero for any real value of x, as can be seen below in the graph of the quadratic polynomial  -x² + x - 1 

 





Thus, we can say that when D< 0, the sign of a quadratic poynomial y  is same as the sign of ''a''. When ''a'' is +ve, y is always +ve, and when ''a'' is -ve, y is always -ve. Note that ''a''cannot be zero, otherwise y would not be a quadratic polynomial.
Cracking IIT just got more exciting,It s not just all about getting assistance from IITians, alongside Target Achievement and Rewards play an important role. ASKIITIANS has it all for you, wherein you get assistance only from IITians for your preparation and win by answering queries in the discussion forums. Reward points 5 + 15 for all those who upload their pic and download the ASKIITIANS Toolbar, just a simple  to download the toolbar….
So start the brain storming…. become a leader with Elite Expert League ASKIITIANS
Thanks
Aman Bansal
Askiitian Expert