Multiple choice

What will be the value of $k$ for which the number $3$ lies between the roots of the below mentioned equation? $x^2 + (1 - 2k) x + (k^2 - k - 2) = 0$.

  1. $ k \in (-\infty,2) \cup (5,\infty) $
  2. $ k = \cfrac {1}{2} $
  3. $k \in (2,5)$
  4. $ k \in (-1,2) $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For 3 to lie between roots of f(x)=x^2+(1-2k)x+(k^2-k-2), f(3) must be < 0. f(3) = 9 + 3(1-2k) + k^2-k-2 = 9 + 3 - 6k + k^2 - k - 2 = k^2 - 7k + 10 < 0. (k-2)(k-5) < 0. So k is in (2, 5).

AI explanation

For the number 3 to lie between the roots of a quadratic equation, the value of the polynomial evaluated at x equals 3 must be less than zero, since the coefficient of x squared is positive. Substituting x equals 3 into x squared plus the quantity 1 minus 2 k times x plus the quantity k squared minus k minus 2 gives 9 plus 3 minus 6 k plus k squared minus k minus 2, which simplifies to k squared minus 7 k plus 10. Factoring this quadratic gives k minus 2 times k minus 5 is less than 0, which holds true when k lies strictly between 2 and 5. Therefore, the required range for k is the interval from 2 to 5.

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