Multiple choice

The roots of the equation $(c^2- ab)x^2 - 2(a^2 -bc)x + (b^2 - ac) =0$ are equal then

  1. $a^3 + b^3 + c^3 = 3abc$ or $a = 0$
  2. $a + b + c = 0$
  3. $a^3 + b^3 + c^3 = 3abc$ or $a = 1$

  4. none of these
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A Correct answer
Explanation

For a quadratic equation Ax^2 + Bx + C = 0 to have equal roots, the discriminant D = B^2 - 4AC must be 0. Substituting the given coefficients and simplifying leads to the condition a^3 + b^3 + c^3 = 3abc or a = 0.

AI explanation

For the roots of the quadratic equation (c^2 - ab)x^2 - 2(a^2 - bc)x + (b^2 - ac) = 0 to be equal, the discriminant must be zero, giving 4(a^2 - bc)^2 - 4(c^2 - ab)(b^2 - ac) = 0. Factoring the resulting expression yields a(a^3 + b^3 + c^3 - 3abc) = 0. This equality holds true if either a = 0 or a^3 + b^3 + c^3 = 3abc.

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