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Quadratic Equation

If the equation is in the form of ax2 + bx + c= 0 (where a ¹ 0). Then this known as quadratic equation or equation of degree 2.

 

If the quadratic equation is ax2 + bx + c = 0
 

Then the values of x are called roots

 

and,       
 

i.e.          
 

 

Thus, the roots of the equation are :
 

 


 

Symbolically, the roots can be expressed as –
 

 

 

Relation  between Roots

 Let one root be  and the other root be b of the equation ax2 + bx + c = 0. Then

 

              
 

              Product of the roots  =
 

 

 



  In order to find related µ and b, the following results may be useful :
 

              (i)   µ2 + b2 = (µ + b)2 - 2µ b
 

              (ii)  (µ - b)2 = (µ + b)2 - 4µ b
 

              (iii) µ3 + b3 = (µ + b)3 - 3µ b (µ + b)
 

              (iv) µ3 - b3  = (µ - b)3 + 3µ b (µ - b)

 

Construction of a quadratic equation 

 

If µ and b be the roots of the equation. Then the quadratic equation are :
 

                                 
 

    i.e.       x2 – (sum of the roots) x + Product of the roots = 0

 

Nature of the roots 

 

1.  If b2 – 4ac (discriminant) = 0; roots are real and equal
 

2.  If b2 – 4ac < 0; roots are imaginary
 

3.  If b2 – 4ac > 0 and a perfect square. Roots are real, rational and unequal.
 

4.  If b2 – 4ac > 0 but not a perfect square. Roots are real, irrational and unequal.

 

       

 





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