QUADRATIC EQUATION - Concept, Properties with Example

QUADRATIC EQUATION

 

INTRODUCTION

The algebraic expression of the form ax2 + bx + c, a 0 is called a quadratic expression, because the highest order term in it is of second degree. Quadratic equation means, ax2 + bx + c = 0. In general whenever one says zeroes of the expression ax2+bx + c, it implies roots of the equation ax2 + bx + c = 0, unless specified otherwise.

A quadratic equation has exactly two roots which may be real (equal or unequal) or imaginary.

SOLUTION OF QUADRATIC EQUATION & RELATION BETWEEN ROOTS & CO-EFFICIENTS

  • The general form of quadratic equation is

          The roots can be found in following manner 

This expression can be directly used to find the two roots of a quadratic equation.

  • The expression b2 – 4 ac = D is called the discriminant of the quadratic equation
  • if are the roots of the quadratic equation then 

         

               

              

  •     

Example : 

 

Solution : 

Nature of Roots :

(a). Consider the quadratic equation ax2 + bx + c = 0 where a,b,c are real number a!=0 then

      

  1. D > 0 => roots are real & distinct (unequal).
  2. D = 0 => roots are real & coincident (equal).
  3. D < 0 => roots are imaginary.
  4. if p + i q is one root of a quadratic equation, then the other root must be the conjugate p–iq&vice versa. 

(b).

 

Example : 

 

Solution : 

 

Example :

Find all the integral values of a for which the quadratic equation (x – a)(x – 10) + 1 = 0 has integral roots.

 

Solution : 

Here the equation is 

Since integral roots will always be rational it means D should be a perfect square.

So 

If D is a perfect square it means we want difference of two perfect square as 4 which is possible only when

(a–10)(a-10) = 4 and D = 0.

so

(a-10) = +2 or (a-10) = -2   

value of a should be 12, 8

 

ROOTS UNDER PARTICULAR CASES