Multiple choice

If $y=\cfrac { 2 }{ 3 } $ is a root of the quadratic equation $3{ y }^{ 2 }-ky+8=0$, then the value of $k$ is ..................

  1. $13$
  2. $-14$
  3. $14$
  4. $-13$
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C Correct answer
Explanation

If y = 2/3 is a root, then 3(2/3)^2 - k(2/3) + 8 = 0. This is 3(4/9) - 2k/3 + 8 = 0, which is 4/3 - 2k/3 + 8 = 0. Multiplying by 3: 4 - 2k + 24 = 0, so 28 = 2k, k = 14.

AI explanation

Substitute the root y = 2/3 directly into the quadratic equation 3y^2 - ky + 8 = 0. This gives the equation 3(2/3)^2 - k(2/3) + 8 = 0, which simplifies to 3(4/9) - 2k/3 + 8 = 0 or 4/3 - 2k/3 + 8 = 0. Multiplying by 3 yields 4 - 2k + 24 = 0, so 2k = 28 and k = 14.

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