If $a = b = c$ then show that the roots of the equation $(x - a) (a - b) + (x - b) (x - c) + (x - c) (x - a)$ are rational, positive and real

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Solution

Given $a = b = c$
value of roots of
$(x - a) (x - b) + (x - b) (x - c) + (x - c) (x - a)$
$\Rightarrow (x - a) (x - a) - (x - a) (x - a) + (x - a) (x - a) (∵ a = b = c)$
$\Rightarrow 3 (x - a)^{2}$
$∴$ the roots are rational and positive and equal.

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If a=b=c then show that the roots of the equation (x−a)(a−b)+(x−b)(x−c)+(x−c)(x−a) are rational, positive and real

Given a=b=cvalue of roots of(x−a)(x−b)+(x−b)(x−c)+(x−c)(x−a)⇒ (x−a)(x−a)−(x−a)(x−a)+(x−a)(x−a) (∵ a=b=c)⇒ 3(x−a)2∴ the roots are rational and positive and equal.

If $a = b = c$ then show that the roots of the equation $(x - a) (a - b) + (x - b) (x - c) + (x - c) (x - a)$ are rational, positive and real

Given $a = b = c$
value of roots of
$(x - a) (x - b) + (x - b) (x - c) + (x - c) (x - a)$
$\Rightarrow (x - a) (x - a) - (x - a) (x - a) + (x - a) (x - a) (∵ a = b = c)$
$\Rightarrow 3 (x - a)^{2}$
$∴$ the roots are rational and positive and equal.