If the roots of the equation $(b - c) x^{2} + (c - a) x + (a - b) = 0$ are equal then prove that 2b = a + c

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Solution

If the roots of the given equation are equal then discriminant is zero i. e.
${(c - a)}^{2} - 4 (b - c) (a - b) = 0$
$\Rightarrow c^{2} + a^{2} - 2 a c + 4 b^{2} - 4 a b + 4 a c - 4 b c = 0$
$\Rightarrow c^{2} + a^{2} + 4 b^{2} + 2 a c - 4 a b - 4 b c = 0$
$\Rightarrow {(c + a - 2 b)}^{2} = 0$
$\Rightarrow c + a = 2 b$

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