Discuss the nature of the roots of the equations
(a) $(a + c - b) x^{2} + 2 c x + (b + c - a) = 0$
(b) $(b + c) x^{2} - (a + b + c) x + a = 0$
(c) $2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0$

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Solution

(a) $B^{2} - 4 A C = 4 {(a - b)}^{2} = + i v e$ and perfect square.
$∴$ Roots are real, rational and unequal.
(b) $B^{2} - 4 A C = {(b + c - a)}^{2} \geq 0$ .
Roots are real, rational and unequal.
(c) $B^{2} - 4 A C = - 4 {(a - b)}^{2} = - i v e$ .
Hence roots are imaginary.

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