Notes – Quadratics

Roots of the equation  ax2+bx+c=0

Multiplying the equation  ax2+bx+c=0   both sides by  4a , we get


Adding  b2  to both sides




x=b ± b24ac2a

Sum and Product of the roots:

If  α  and  β  are the two roots of the above equation, then

α=b+b24ac2a,    β=b±b24ac2a

Adding the above two, 

Sum of the roots =  α+β=b+b24ac bb24ac2a =2b2a=ba

Product of the roots:

αβ=b2b24ac24a2   =b2b2+4ac4a2=4ac4a2=ca

Nature of the roots:

Transformation of equations:

Let  α and  β  are the roots of the equation  ax2+bx+c=0

To find the equation whose roots are :

(i) Negative of the roots of the equation  ax2+bx+c=0

The required roots are  α   and   β .

This can be obtained by substituting

y=α=x      x=y

so,  ay2+by+c=0   ay2by+c=0

or       ax2+bx+c=0

(ii) Increased by  h  i.e.  α+h , β+h

substituting  y=α+h=x+h   x=yh

so,  ayh2+byh+c=0


so the required equation is 


Sign of coefficient determining the sign of both real roots of ax2+bx+c=0