Multiple choice

The difference between the two roots of a quadratic equation is $2$ and the difference between the cubes of the roots is $98$, then which of the following is that quadratic equation?

  1. ${x}^{2}-8x+15=0$
  2. ${x}^{2}+8x-15=0$
  3. ${x}^{2}+5x+15=0$
  4. ${x}^{2}-5x-15=0$
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A Correct answer
Explanation

Let roots be a, b. a-b=2 and a^3-b^3=98. Since a^3-b^3 = (a-b)(a^2+ab+b^2) = 98, we have 2(a^2+ab+b^2)=98, so a^2+ab+b^2=49. Using (a-b)^2 = a^2-2ab+b^2=4, we find 3ab = 45, so ab=15. The equation is x^2 - (a+b)x + ab = 0. Since (a+b)^2 = (a-b)^2 + 4ab = 4 + 60 = 64, a+b=8. Thus x^2-8x+15=0.

AI explanation

Let the roots be alpha and beta, where alpha - beta = 2. The difference of their cubes can be factored as (alpha - beta)(alpha^2 + alpha*beta + beta^2) = 98. Substituting the known difference gives 2((alpha + beta)^2 - alpha*beta) = 98, which simplifies to (alpha + beta)^2 - alpha*beta = 49. By expressing this in terms of the sum S = alpha + beta and product P = alpha*beta, we have S^2 - P = 49. We know (alpha - beta)^2 = S^2 - 4P, so 4 = S^2 - 4P. Solving the system S^2 - P = 49 and S^2 - 4P = 4 yields 3P = 45, making P = 15. Substituting P back gives S^2 = 64, so S = 8 or -8. The quadratic equation is x^2 - Sx + P = 0, which gives x^2 - 8x + 15 = 0.

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