Multiple choice

If $2$ and $3$ are the roots of the equation $2 x ^ { 3 } + m x ^ { 2 } - 13 x + n = 0 ,$ the third root of this equation is

  1. $-5$
  2. $30$
  3. $- \dfrac { 5 } { 2 }$
  4. $- 15$
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C Correct answer
Explanation

Let the roots be 2, 3, and r. By Vieta's formulas, the sum of roots 2 + 3 + r = -m/2, and the product of roots 2 * 3 * r = -n/2. Also, the sum of roots taken two at a time is 2*3 + 2*r + 3*r = -13/2. So, 6 + 5r = -6.5, which means 5r = -12.5, so r = -2.5.

AI explanation

Let the third root of the cubic equation be k, so the sum of the roots is 2 + 3 + k. Using Vieta's formulas, the sum of the roots is also equal to -m/2, which establishes the relation 2 + 3 + k = -m/2. We substitute the root x = 2 into the equation to get 16 + 4m - 26 + n = 0, simplifying to 4m + n = 10, and substitute x = 3 to get 54 + 9m - 39 + n = 0, simplifying to 9m + n = -15. Solving this system of equations gives m = -5 and n = 30, which we plug back into the sum formula to find 5 + k = 2.5, resulting in k = -5/2.

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