Multiple choice

If $a + b + c = 0$ and $a, b, c$ are rational, then the roots of the equation $(b + c - a)x^2 + (c + a - b)x + (a + b - c) = 0$ are rational .

  1. Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I .

  2. Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-I .

  3. Statement-I is true, Statement-II is false .

  4. Statement-I is false, Statement-II is true .
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A Correct answer
AI explanation

Because a + b + c = 0, we can substitute b + c = -a, c + a = -b, and a + b = -c into the given equation. The equation simplifies to -2ax^2 - 2bx - 2c = 0, or ax^2 + bx + c = 0. The sum of the roots is -b/a, the product is c/a, and the discriminant is b^2 - 4ac. Using the condition a + b + c = 0, we replace b^2 with (a + c)^2, making the discriminant (a + c)^2 - 4ac = (a - c)^2. Since a and c are rational, the discriminant is a perfect square of a rational number, meaning the roots are rational; thus, both statements are true and the second correctly explains the first.

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