Multiple choice

If the equation $\displaystyle x^{2}-2kx-2x+k^{2}=0 $ has equal roots the value of k must be

  1. zero
  2. either zero or $\displaystyle -\frac{1}{2} $
  3. $\displaystyle -\frac{1}{2} $
  4. either $\displaystyle \frac{1}{2} $ or $\displaystyle -\frac{1}{2} $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For x^2 - 2(k+1)x + k^2 = 0 to have equal roots, the discriminant must be 0. D = (2(k+1))^2 - 4(1)(k^2) = 4(k^2 + 2k + 1) - 4k^2 = 8k + 4. Setting 8k + 4 = 0 gives k = -1/2.

AI explanation

A quadratic equation has equal roots when its discriminant, b^2 - 4ac, is equal to zero. For the equation x^2 - 2kx - 2x + k^2 = 0, group the terms to get x^2 - (2k + 2)x + k^2 = 0. Applying the discriminant formula gives (2k + 2)^2 - 4(1)(k^2) = 0. Expanding yields 4k^2 + 8k + 4 - 4k^2 = 0, which simplifies to 8k = -4, meaning k is -1/2.

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