Multiple choice

If $\lambda \epsilon R$ is such that the sum of the cubes of the roots of the equation, $x^{2} + (2 - \lambda) x + (10 - \lambda) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is

  1. $20$
  2. $2\sqrt {5}$
  3. $2\sqrt {7}$
  4. $4\sqrt {2}$
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B Correct answer
Explanation

Let the roots be a and b. a+b = lambda-2 and ab = 10-lambda. Sum of cubes a^3+b^3 = (a+b)^3 - 3ab(a+b). Substituting the expressions, we minimize the function of lambda. The difference of roots is sqrt((a+b)^2 - 4ab).

AI explanation

Let the roots be p and q, where the sum p + q = L - 2 and the product pq = 10 - L. Using the identity for the sum of cubes, p^3 + q^3 = (p + q)^3 - 3pq(p + q), we substitute the values to get (L - 2)^3 - 3(10 - L)(L - 2). Expanding and simplifying this expression yields the cubic polynomial L^3 - 9L^2 + 24L - 8, whose derivative is 3L^2 - 18L + 24. Setting the derivative to zero gives L = 2 or L = 4, and testing these values reveals the minimum sum of cubes occurs at L = 4. Plugging L = 4 back into the original quadratic gives x^2 - 2x + 6 = 0, where the magnitude of the difference of the roots is the square root of the discriminant, resulting in 2 * sqrt(5).

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