Solutions to worksheet





             In this page 'Solutions to worksheet' we are going to see how to solve the quadratic equations with complex roots given in the page 'Worksheet on 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.

Solutions to worksheet

Find the roots of the given quadratic equations:

1.  x² - x + 1 = 0

Solution:

    Here

           a = 1, b = -1, c =1

The value of x is

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

                                              2(1)

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

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

                       =   (1 ± √3i)/2

                 x   =  (1 ± √3i)/2

So, A  is the answer.


2. Find the roots of the given quadratic equation

                        x² + 3x + 5 = 0

Solution:

          Here a = 1, b = 3, c =5

      The value of x is

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

                                              2(1)

                    x  =   [-3 ± √(9-20)]/2

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

                       =   (-3 ± √11i)/2

                 x   =  (-3 ± √11i)/2

So B is the answer.


3. find the roots of the given quadratic equation

                      -5/x = x-2

Solution:

Re writing the given equation, 

              x (-5/x)   =   x(x-2)

                   -5      =   x² - 2x

          x²-2x + 5    =   0

        Here a = 1, b = -2, c =5

 The value of x is

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

                                              2(1)

                    x  =   [2 ± √(4-20)]/2

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

                       =   (2 ± 4i)/2

                 x   =  1 ± 4i

So C is the answer.


4. Solve the given quadratic equation

                       3x²+ 10x + 9 =0

Solution:

    Here a = 3,  b = 10, c =9

The value of x is

                     x  =  -(10) ± √[(10)² - 4(3)(9)]

                                              2(3)

                    x  =   [-10 ± √(100-108)]/6

                        =  [-10 ± √(-8)]/6

                       =   (-5± √2i)/3

                 x   = ( -5 ± √2i)/3

So  A is the answer.


5. Find the roots of the quadratic equation

            (-x+3) = 2/(x-2)

Solution:

Re writing the given equation

        (x-2)(-x+3) = [2/(x-2)](x-2)

       - x²+ 2x +3x-6 = 2

          x² - 5x + 8 = 0

      Here a = 1, b = -5, c = 8

The value of x is

                     x  =  -(-5) ± √[(-5)² - 4(1)(8)]

                                              2(1)

                    x  =   [5 ± √(25-32)]/2

                        =  [5 ± √(-7)]/2

                       =   (5± √7i)/2

                 x   = ( 5 ± √7i)/2

So C is the answer.


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