Find the nature of the roots …

For a quadratic equation ax² + bx + c =0, the term b² - 4ac is called discriminant (D) of the quadratic equation because it  determines  whether the quadratic equation has real roots or not ( nature of roots).

D=  b² - 4ac

So a quadratic equation ax² + bx + c =0, has

i) Two distinct real roots, if b² - 4ac >0 , then x= -b/2a + √D/2a  &x= -b/2a - √D/2a

ii) Two equal real roots, if b² - 4ac = 0 , then x= -b/2a or -b/2a

iii) No real roots, if b² - 4ac <0

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Solution:

i)

x² – 3x + 5 = 0
 

Comparing it with ax² + bx + c = 0, we get
 

a = 2, b = -3 and c = 5
 

Discriminant (D) = b² – 4ac
 

⇒ ( – 3)2 – 4 (2) (5) = 9 – 40
⇒ – 31<0

As b2 – 4ac < 0,
 

Hence, no real root is possible .

(ii) 3x² – 4√3x + 4 = 0
 

Comparing it with ax² + bx + c = 0, we get
 

a = 3, b = -4√3 and c = 4
 

Discriminant(D) = b² – 4ac
 

⇒ (-4√3)2 – 4(3)(4)
 

⇒ 48 – 48 = 0
 

As b² – 4ac = 0,
Hence,  real roots exist & they are equal to each other.
 

the roots will be –b/2a and –b/2a.

-b/2a = -(-4√3)/2×3 = 4√3/6 = 2√3/3 

multiplying the numerator & denominator by √3

(2√3) (√3) / (3)(√3)  = 2 ×3 / 3 ×√3 = 2/√3

Hence , the equal roots are 2/√3 and 2/√3.