Find the nature of the roots of the quadratic equation $(b - c) x^{2} + 2 (c - a) x + (a - b) = 0$ .

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Solution

$(b - c) x^{2} + 2 (c - a) x + (a - b) = 0$

$D = b^{2} - 4 a c = (c - a)^{2} - (b - c) (a - b)$

$c^{2} + a^{2} - 2 a c - a b + b^{2} + a c - b c$

$= a^{2} + b^{2} + c^{2} - a c - b c - a b$

$= 2 (a^{2} + b^{2} + c^{2} - a b - b c - c a)$

So, the roots are real and may or may not be equal.

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