Multiple choice

Determine the values of $m$ for which the equation $\displaystyle 5x^{2}-4x+2+m\left ( 4x^{2}-2x-1 \right )= 0$ will have p roduct of roots as $2$.

  1. $-\dfrac89$
  2. $-\dfrac79$
  3. $-\dfrac67$
  4. $\dfrac79$
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A Correct answer
Explanation

For the quadratic, the product of roots is c/a = (2 - m)/(5 + 4m). Setting this equal to 2 gives 2 - m = 2(5 + 4m), so -9m = 8 and m = -8/9.

AI explanation

Rearrange the equation to standard quadratic form: (5+4m)x^2 + (-4-2m)x + (2-m) = 0. Using the product of roots formula c/a = 2, we set (2-m)/(5+4m) = 2. Solving this equation gives 2 - m = 10 + 8m, which simplifies to 9m = -8, so m = -8/9.

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