Multiple choice

If $ { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }-3abc=0$ then the roots of the equation $ \left( { a }^{ 2 }-bc \right) { x }^{ 2 }+2\left( { b }^{ 2 }-ac \right) x+{ c }^{ 2 }-ab=0$ are

  1. imaginary

  2. real and unequal

  3. real and equal

  4. Cannot say
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The condition a^3 + b^3 + c^3 - 3abc = 0 implies either a+b+c = 0 or a=b=c. If a=b=c, the quadratic equation becomes (a^2-a^2)x^2 + 2(a^2-a^2)x + (a^2-a^2) = 0, which is 0=0, but more generally, the discriminant of the given quadratic equation simplifies to zero under these conditions, leading to real and equal roots.

AI explanation

The identity a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) implies that a^2 + b^2 + c^2 = ab + bc + ca. For the given quadratic, the discriminant evaluates to [2(b^2 - ac)]^2 - 4(a^2 - bc)(c^2 - ab), which simplifies to 4(ab + bc + ca)^2 - 4(a^2 + b^2 + c^2)^2. Substituting the equality makes the discriminant zero, meaning the roots are real and equal.

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