Multiple choice

If the product of the roots of the equation $ 5x^{2}-4x+2+k(4x^{2}-2x-1) =0$ is $2$, then $k =$

  1. $-\dfrac{8}{9}$
  2. $\dfrac{8}{9}$
  3. $\dfrac{4}{9}$
  4. $-\dfrac{4}{9}$
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A Correct answer
Explanation

Equation: (5+4k)x^2 - (4+2k)x + (2-k) = 0. Product of roots = c/a = (2-k)/(5+4k) = 2. 2-k = 10 + 8k. -8 = 9k. k = -8/9.

AI explanation

Group the terms to form a single quadratic equation: (5 + 4k)x^2 + (-4 - 2k)x + (2 - k) = 0. The product of the roots is the constant term divided by the leading coefficient, so (2 - k) / (5 + 4k) equals 2. Solving this equation gives 2 - k = 10 + 8k, which simplifies to -9k = 8. Thus, k equals -8/9.

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