Multiple choice

The roots of the equation $x^{2}+6x+a=0$ are real and distinct and they differ by atmost $4$, then find the range of values of $a.$

  1. $(5, 9]$
  2. $[5, 9)$
  3. $[5, \infty)$
  4. $[3, 9)$
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B Correct answer
Explanation

For real and distinct roots, the discriminant D = 36 - 4a must be > 0, so a < 9. If the roots are r1 and r2, then |r1 - r2| = sqrt(D)/|a_coeff| = sqrt(36 - 4a). Given the difference is at most 4, sqrt(36 - 4a) <= 4 implies 36 - 4a <= 16, so 4a >= 20, or a >= 5. Combining these, 5 <= a < 9.

AI explanation

For the roots to be real and distinct, the discriminant must be positive, so 36 - 4a > 0, meaning a < 9. Let the roots be p and q; we know their sum is -6 and their difference is at most 4, establishing (p-q)^2 <= 16. Since (p-q)^2 equals (p+q)^2 - 4pq, we substitute the known values to get 36 - 4a <= 16, which simplifies to a >= 5. Combining the inequalities from the discriminant and the difference condition gives the range [5, 9).

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