# Nature of Roots

The nature of the roots depend entirely on the discriminant (D) = D = b^{2 }â€“ 4ac.

Case â€“ I If a, b, c are real numbers and a â‰ 0, then if

- b
^{2}â€“ 4ac = 0 â‡’ roots are real and equal - b
^{2 }â€“ 4ac > 0 â‡’ roots are real and unequal - b
^{2 }â€“ 4ac < 0 â‡’ roots are imaginary (also complex conjugates of each other)

Case â€“ II If a, b, c are rational numbers and a â‰ 0, then if

- b
^{2 }â€“^{ }4ac = 0 â‡’ roots are rational and equal - b
^{2 }â€“^{ }4ac > 0 and b^{2}â€“ 4ac is a perfect square â‡’ roots are rational and unequal - b
^{2 }â€“ 4ac > 0 and b^{2}â€“ 4ac is not perfect square â‡’ roots are irrational and unequal - b
^{2 }â€“ 4ac < 0 â‡’ the roots are imaginary (also complex conjugates of each other)

- If
*a*and*b*are the roots of equation - Also, for roots
*a*and*b*the quadratic equation is

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