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Nature of Roots

The nature of the roots depend entirely on the discriminant (D) = D = b2 – 4ac.

Case – I If a, b, c are real numbers and a ≠ 0, then if

  1. b2 – 4ac = 0  roots are real and equal
  2. b2 – 4ac > 0  roots are real and unequal
  3. b2 – 4ac < 0  roots are imaginary (also complex conjugates of each other)

Case – II If a, b, c are rational numbers and a ≠ 0, then if

  1. b2  4ac = 0  roots are rational and equal
  2. b2  4ac > 0 and b2 – 4ac is a perfect square  roots are rational and unequal
  3. b2 – 4ac > 0 and b2 – 4ac is not perfect square  roots are irrational and unequal
  4. b2 – 4ac < 0  the roots are imaginary (also complex conjugates of each other)
  • If a and b are the roots of equation Description: 37955.png
     
    So, sum of roots =Description: 37949.png
  • Also, for roots a and b the quadratic equation is
     
    Description: 37943.png 





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