Multiple choice

The complete set of values of $k$, for which the quadratic equation ${x}^{2}-kx+k+2=0$ has equal roots, consists of

  1. $2+\sqrt{12}$
  2. $2\pm \sqrt{12}$
  3. $2-\sqrt{12}$
  4. $-2-\sqrt{12}$
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B Correct answer
Explanation

For a quadratic equation to have equal roots, the discriminant (b^2 - 4ac) must be zero. For x^2 - kx + (k+2) = 0, the discriminant is (-k)^2 - 4(1)(k+2) = k^2 - 4k - 8 = 0. Using the quadratic formula, k = (4 +/- sqrt(16 - 4(1)(-8))) / 2 = (4 +/- sqrt(48)) / 2 = 2 +/- sqrt(12).

AI explanation

Using the condition for equal roots, we set the discriminant equal to zero. For x^2 - kx + k + 2 = 0, we have b^2 - 4ac = (-k)^2 - 4(1)(k + 2) = 0. Solving the quadratic equation k^2 - 4k - 8 = 0 using the quadratic formula yields k = 2 plus or minus the square root of 12.

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