Multiple choice

The roots of the equation ${x}^{2}-\cfrac{2}{3}x=32$ are

  1. $6,\cfrac{-15}{4}$
  2. $-6,\cfrac{-16}{3}$
  3. $3,16$
  4. $\cfrac{-16}{3},6$
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D Correct answer
Explanation

x^2 - (2/3)x - 32 = 0. Multiply by 3: 3x^2 - 2x - 96 = 0. Using quadratic formula: x = [2 +/- sqrt(4 - 4*3*(-96))] / 6 = [2 +/- sqrt(4 + 1152)] / 6 = [2 +/- sqrt(1156)] / 6 = [2 +/- 34] / 6. Roots are 36/6 = 6 and -32/6 = -16/3.

AI explanation

To eliminate the fraction, multiply the entire equation by 3 to get 3x^2 - 2x = 96, which rearranges to 3x^2 - 2x - 96 = 0. Splitting the middle term, we get 3x^2 - 18x + 16x - 96 = 0. Factoring by grouping gives 3x(x - 6) + 16(x - 6) = 0, or (3x + 16)(x - 6) = 0. Setting each factor to zero gives x = -16/3 and x = 6.

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