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 s um of the roots as $6$.

  1. $-\dfrac{11}{15}$
  2. $-\dfrac{13}{15}$
  3. $-\dfrac{13}{11}$
  4. $-\dfrac{15}{13}$
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C Correct answer
Explanation

Combining terms gives (5 + 4m)x^2 + (-4 - 2m)x + (2 - m) = 0. The sum of roots is (4 + 2m)/(5 + 4m), so setting it equal to 6 gives m = -13/11.

AI explanation

Rearrange the equation into standard quadratic form: (5+4m)x^2 + (-4-2m)x + (2-m) = 0. Using the sum of roots formula -b/a = 6, we set (4+2m)/(5+4m) = 6. Solving this equation gives 4 + 2m = 30 + 24m, which simplifies to -22m = 26, so m = -13/11.

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