Multiple choice

If a, b, c $\epsilon\ Q\ $, then the roots of the equation $(b + c - 2a) x^{2} + (c+a-2b) x+ (a+b-2c) = 0$ are

  1. Rational

  2. Irrational

  3. Integers

  4. Complex
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A Correct answer
Explanation

The discriminant of the quadratic equation Ax^2 + Bx + C = 0 is D = B^2 - 4AC. Substituting the given coefficients, D = (c+a-2b)^2 - 4(b+c-2a)(a+b-2c). Simplifying this expression results in a perfect square, (3a-3c)^2, which implies the roots are rational since a, b, and c are rational.

AI explanation

Add the three coefficients (b plus c minus 2a), (c plus a minus 2b), and (a plus b minus 2c) to get a sum of 0. This means that x equals 1 is a root of the quadratic equation. Since the product of the roots is (a plus b minus 2c) divided by (b plus c minus 2a), the other root must be (a plus b minus 2c) divided by (b plus c minus 2a). Because a, b, and c are rational numbers, both roots are rational. The roots are therefore rational.

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