Multiple choice

The roots of the equation $(\mathrm{a}+\mathrm{c}-\mathrm{b})x^{2}+2cx+(\mathrm{b}+\mathrm{c}-\mathrm{a})=0$, where $(a, b, c)\in R$ and $(a\neq b)$ are

  1. real and distinct

  2. real and equal

  3. real

  4. imaginary
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A Correct answer
Explanation

The discriminant D = (2c)^2 - 4(a+c-b)(b+c-a) = 4c^2 - 4((c+(a-b))(c-(a-b))) = 4c^2 - 4(c^2 - (a-b)^2) = 4(a-b)^2. Since a != b, D > 0, so the roots are real and distinct.

AI explanation

To determine the nature of the roots, we calculate the discriminant D, which equals B squared minus 4AC for the equation (a+c-b)x squared plus 2cx plus (b+c-a) equals 0. Substituting the coefficients gives D equals (2c) squared minus 4(a+c-b)(b+c-a), which simplifies to 4c squared minus 4(c squared - (a-b) squared). This expression evaluates to 4(a-b) squared, and since a does not equal b, the discriminant is strictly greater than zero. Therefore, the roots of the given equation are real and distinct.

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