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.
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