## 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.

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

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

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

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

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