Multiple choice

If both the roots of the equation $x^2+(3-2k)x-6k=0$ belong to the interval $(-6, 10)$, then the largest value of the k is.

  1. $-1$
  2. $3$
  3. $5$

  4. Not defined
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C Correct answer
AI explanation

The roots of x^2 + (3-2k)x - 6k = 0 must lie within the interval (-6, 10), meaning the quadratic evaluated at the boundaries must be positive: f(-6) > 0 and f(10) > 0. Calculating f(-10) gives 100 + 30 - 20k - 6k > 0, so 130 > 26k, which means k < 5. Calculating f(-6) gives 36 - 18 + 12k - 6k > 0, so 18 > -6k, meaning k > -3. Combining these inequalities restricts k to the range (-3, 5), making the largest integer value 5.

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