Multiple choice

If the roots of the quadratic equation $x^2 + 6x + b = 0$ are real and distinct and they differ by atmost $4$, then the least value of $b$ is

  1. $5$
  2. $6$
  3. $7$
  4. $8$
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A Correct answer
Explanation

Roots differ by at most 4: |x1 - x2| <= 4. |x1 - x2| = sqrt(D)/|a| = sqrt(36 - 4b) <= 4. Square both sides: 36 - 4b <= 16. 20 <= 4b, so b >= 5. For real and distinct roots, D > 0, so 36 - 4b > 0, b < 9. The least value is 5.

AI explanation

For the quadratic equation x^2 + 6x + b = 0, the sum of the roots is -6 and the product is b. If the roots differ by at most 4, we have (alpha - beta)^2 <= 16. Expanding this gives (alpha + beta)^2 - 4*alpha*beta <= 16, so (-6)^2 - 4b <= 16. This simplifies to 36 - 4b <= 16, which means 4b >= 20 and b >= 5. The least value of b is 5.

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