Multiple choice

If $ \alpha, \beta $ are the roots of equation $(k + 1) x^{2} - (20k + 14)x + 91k + 40 = 0 ; (\alpha < \beta)$, $k > 0$, then The nature of the roots of this equation is

  1. imaginary

  2. real and distinct

  3. equal real roots

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Discriminant D = (20k+14)^2 - 4(k+1)(91k+40) = 400k^2 + 560k + 196 - 4(91k^2 + 131k + 40) = 400k^2 + 560k + 196 - 364k^2 - 524k - 160 = 36k^2 + 36k + 36. Since 36(k^2+k+1) > 0 for all real k, the roots are real and distinct.

AI explanation

To determine the nature of the roots, we calculate the discriminant, D = b^2 - 4ac. Substituting the coefficients gives D = [-(20k + 14)]^2 - 4(k + 1)(91k + 40) = 400k^2 + 560k + 196 - (364k^2 + 524k + 160). This simplifies to D = 36k^2 + 36k + 36 = 36(k^2 + k + 1), which is strictly positive for all real values of k. Therefore, the roots are real and distinct.

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