Multiple choice

Comment upon the nature of roots of the quadratic equation $\displaystyle x^{2}+2x=k+\int_{0}^{1}\left | t+k \right |dx$ depending on the value of $\displaystyle k\in R$

  1. real & distinct $\displaystyle \forall k\in R$
  2. only distinct $\displaystyle \forall k\notin R$
  3. only real $\displaystyle \forall k\in R$

  4. none of above
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A Correct answer
Explanation

The discriminant is 4(1 + k + integral from 0 to 1 of |t + k| dt), which remains positive for every real k. Therefore, the quadratic always has two real and distinct roots.

AI explanation

Rewrite the equation as x^2 + 2x - (k + integral) = 0. The nature of the roots depends on the discriminant, D = b^2 - 4ac, where a = 1, b = 2, and c = -(k + integral). Calculating the discriminant gives D = (2)^2 - 4(1)(-(k + integral)) = 4 + 4(k + integral) = 4(1 + k + integral). Since the definite integral of |t + k| from 0 to 1 is always greater than or equal to -|1 + k| - |k|, the sum of 1, k, and the integral is always strictly greater than zero. Therefore, D is strictly positive for all real k, meaning the roots are real and distinct.

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