Multiple choice

For the equation $x^2 - (k + 1)x + (k^2 + k - 8) = 0$ if one root is greater then $2$ and other is less than $2$, then $k$ lies between

  1. $-2$ & $3$
  2. $2$ & $-2$
  3. $2$ & $-3$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a quadratic f(x) = x^2 - (k+1)x + (k^2+k-8) to have one root < 2 and one root > 2, f(2) must be less than 0. f(2) = 4 - 2(k+1) + k^2 + k - 8 = k^2 - k - 6 < 0. Factoring gives (k-3)(k+2) < 0, which holds for -2 < k < 3.

AI explanation

Using the condition for a quadratic equation to have one root less than 2 and the other greater than 2, we substitute x = 2 into the expression and require it to be strictly negative. Plugging in 2 gives 2^2 - (k + 1)2 + (k^2 + k - 8), which simplifies to k^2 - k - 6. For the roots to lie on opposite sides of 2, we need k^2 - k - 6 < 0. Factoring the quadratic inequality yields (k - 3)(k + 2) < 0, meaning k lies between -2 and 3.

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