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Prove that the roots of the equation (b-c)x^2 + (c-a)x + (a-b) = 0 are rational given that a,b,c are rational. Find the condition for roots to be equal.

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Answer:

the roots of the equation are rational when a, b, and c are rational, and the roots are equal when b = c.

Explanation:

To prove that the roots of the equation (b-c)x^2 + (c-a)x + (a-b) = 0 are rational given that a, b, and c are rational, and to find the condition for the roots to be equal, we use the discriminant of a quadratic equation. The discriminant (D) should be a perfect square for the roots to be rational.

By calculating the discriminant for the given equation, we obtain D = 3c^2 - 8bc + 3b^2. Since a, b, and c are rational, D is rational as well.

The condition for the roots to be equal is b = c, which we deduce from the quadratic equation and simplifying the discriminant equation. This conclusion is reached through algebraic manipulation and analysis of the expressions.

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