Multiple choice

The value (s) of $k$ for which the quadratic equation $\displaystyle kx^{2}-kx+1=0$ has equal roots is

  1. $k = 0$ only
  2. $k = 4$ only
  3. $k = 0, 4$
  4. $k = -4$
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B Correct answer
Explanation

For a quadratic equation ax^2 + bx + c = 0 to have equal roots, the discriminant D = b^2 - 4ac must be zero. Here, (-k)^2 - 4(k)(1) = 0, so k^2 - 4k = 0. This gives k(k-4) = 0, so k=0 or k=4. Since k=0 makes it not a quadratic, k=4 is the valid solution.

AI explanation

For a quadratic equation to have equal roots, its discriminant, given by b^2 - 4ac, must equal zero. In the equation kx^2 - kx + 1 = 0, the discriminant is (-k)^2 - 4(k)(1), simplifying to k^2 - 4k. Setting the discriminant to zero gives k^2 - 4k = 0, which factors to k(k - 4) = 0. This yields k = 0 or k = 4, but because the coefficient of x^2 cannot be zero in a valid quadratic equation, k cannot be 0, leaving k = 4 only.

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