Multiple choice

If the sum of the roots of a quadratic equation is $2$ and the sum of their cubes is $98$, then find the equation.

  1. $x^2 - 2x + 12 = 0$
  2. $x^2 - 2x - 15 = 0$
  3. $x^2 - 2x + 9 = 0$
  4. $x^2 - 2x + 13 = 0$
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B Correct answer
Explanation

Let roots be a, b. a+b = 2. a^3+b^3 = 98. a^3+b^3 = (a+b)(a^2-ab+b^2) = (a+b)((a+b)^2 - 3ab). 98 = 2(4 - 3ab). 49 = 4 - 3ab. 3ab = -45. ab = -15. Equation: x^2 - (sum)x + (product) = 0 => x^2 - 2x - 15 = 0.

AI explanation

Let the roots be a and b; given a+b = 2, the sum of their cubes a^3+b^3 equals (a+b)^3 - 3ab(a+b) = 98. Substituting the known sum gives 2^3 - 3ab(2) = 8 - 6ab = 98, which solves to ab = -15. Using the standard formula for a quadratic equation, x^2 - (sum)x + (product) = 0, the equation is x^2 - 2x - 15 = 0.

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