Multiple choice

All possible values of $a$, so that $6$ lies between the roots of the equation $x^2 + 2(a - 3)x + 9 = 0$ is

  1. $(- \infty, -2) \cup (2, \infty)$
  2. $\left(- \infty, -\dfrac{3}{4}\right)$
  3. $(2, \infty)$

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

For 6 to lie between the roots of f(x) = x^2 + 2(a-3)x + 9 = 0, we need f(6) < 0. f(6) = 36 + 12(a-3) + 9 = 36 + 12a - 36 + 9 = 12a + 9. 12a + 9 < 0 implies 12a < -9, or a < -3/4.

AI explanation

For the number 6 to lie strictly between the roots of the quadratic equation x^2 + 2(a - 3)x + 9 = 0, the value of the quadratic expression at x = 6 must be strictly negative. Substituting x = 6 gives 6^2 + 2(a - 3)(6) + 9 < 0, which simplifies to 36 + 12a - 36 + 9 < 0. This further reduces to 12a + 9 < 0, or 4a + 3 < 0. Solving this linear inequality yields a < -3/4, meaning a lies in the interval (-infinity, -3/4).

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