Question

find the nature of the roots   { if it exists by quadratic equation}

6x^2 -x-2=0

x^2 -2x+1=0

x^2+x+1=0

2x^2 +5x+5=0

Answer 1

nature of the root can be identified by D=b^2-4ac
If it is -ve then root will imaginary
If it is +ve then root will be real and different 
If it is equal to Zero, then both root will be real and equal
1) 6x^2-x - 2=0
 ax^2+bx+c=o
on comparing , we will find D
1-4*6*-2= -ve then root will be imaginary
2) x^2+x+1=0
D= 1 -4*1*1 = -ve,then rootwill be imaginary
3) 2x^2+5x+5=0
D= 25-4*2*5= -ve, root will be imaginary

Answer 2

We use discriminant for this purpose for the equation ax² + bx + c = 0

D = b² - 4ac

If D > 0 roots are real and distinct 
   D < 0 roots are imaginary and distinct
   D = 0 Roots are real and equal

so 
1)  6x² - x - 2 = 0
  D = (-1²) - 4 x 6 x - 2
  D = 1 + 48 = 49
  So D > 0 roots are real and distinct 

2) x² - 2x + 1 = 0
D = (- 2)² - 4 x 1 x 1 
D = 4 - 4 = 0
So    D = 0 Roots are real and equal

3) x² + x + 1 = 0
 D = 1² - 4 x 1 x 1 = 1 - 4 = - 3 
So  D < 0 roots are imaginary and distinct

4) 2x² + 5x + 5 = 0
  D = 5² - 4 x 5 x 2 = 25 - 40 = - 15 
So D < 0 roots are imaginary and distinct