Complex roots





          Complex roots:  While solving quadratic equations, we get sometimes unreal roots(imaginary roots). 

           While solving quadratic equations, we will find the value of the discriminant to find the nature of the roots.         

       Quadratic equation:  ax² +bx+c=0, a≠0.

   Discriminant ∆= b² - 4 ac

(1) ∆ > 0 but not a perfect square

(2) ∆ > 0 but a perfect square

(3) ∆ = 0

(4) ∆ < 0 but a perfect square

         Nature of roots

Real,unequal and irrational

Real,unequal and rational

Real, equal and rational

Complex roots

Examples:

1.  Solve the equation :

                   x²/2 = 3x - 5

Solution:

      Re writing the quadratic equation

                         x² = 2(3x-5)

                        x² =  6x -10

                        x² - 6x +10 = 0.

                        a = 1,  b = -6, c = 10

        The value of x is

                     x  =  -(-6) ± √[(-6)² - 4(1)(10)]

                                              2(1)

                    x  =   [6 ± √(36-40)]/2

                        =  [6 ± √(-4)]/2

                       =   (6 ± 2i)/2

                       =    3 ± i

      So       x   =   3 + i, or 3-i

2.  Find the roots of the equation:

                  x² - 2 x + 5 = 0

Solution: In this equation

               a = 1, b = -2, c =5

So substituting in the formula

             x  =  -(-2) ± √[(-2)² - 4 (1)(5)]/ 2(1)

                 =    ( 2  ± √ (4-20)) / 2

                 =     ( 2 ± √(-16)) / 2

                 =      (2 ±  4i) / 2

                =        1 ± 2i     

    So   x  =    1+ 2i or 1-2i

3.  Find the roots of the equation:

                    x + 5/x = 3

Solution:

     Rewriting the equation,

                    x(x + 5/x )  = 3x

                   x²  +  5        =  3x

                  x² - 3x + 5   =  0

       Here a = 1,  b = -3, c = 5.   

       So substituting in the formula

             x  =  -(-3) ± √[(-3)² - 4 (1)(5)]/ 2(1)

                 =    ( 23 ± √ (9-20)) / 2

                 =     ( 3 ± √(-11)) / 2

                 =      (3 ± √11 i) / 2

So        x  =    ( 3 + √11 i)/2  or  ( 3 - √11 i)/2


   Students can go through the problems discussed in this page, 'Complex roots' on their own, and can verify the solutions. If you are having any doubt, you can contact us through mail, we will help you to clear your doubts.

   We welcome your valuable suggestions for the betterment of our site. Please use the box given below to express your suggestions.

HTML Comment Box is loading comments...

                                       Home

                              Complex numbers