Worksheet on complex roots





              In this worksheet on complex roots we are going to see problems in 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:

                    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

Worksheet on complex roots

Find the roots of the given quadratic equations:

1.  x² - x + 1 = 0


(A) x = (1 ± i√3)/2

(B) x = 1, 2

(C) x = 1 ± i

2.  x² + 3x + 5 = 0


(A) x+ √2, x-√2

(B) (-3±i√11)/2

(C) -x±i√7

3.  -5/x = x-2


(A) x+ √2, x-√2

(B) (-3±i√11)/2

(C) x±4i

4.  3x²+ 10x + 9 =0


(A) (-5 ± i√2)/3

(B) (-3±i√11)/2

(C) x±4i

5.  (-x+3) = 2/(x-2)


(A) (-5 ± i√2)/3

(B) (-3±i√11)/2

(C) (5±i√7)/2

              Students can try to solve the problems given in 'Worksheet on complex roots' them selves.  They can verify the answers and solutions with the solutions provided. If you are having any doubt you can contact us through mail, we will help you to clear your doubts.

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