Multiple choice

If roots of the equation $x^3-12x^2 + 39x -28=0$ are in A.P., then its common difference is -

  1. $\pm 1$
  2. $\pm 2$
  3. $\pm 3$
  4. $\pm 4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the roots be a-d, a, a+d. The sum of roots is 3a = 12, so a = 4. The product of roots is (4-d)(4)(4+d) = 28. Thus, 16-d^2 = 7, which means d^2 = 9, so d = +/- 3.

AI explanation

Since the roots are in arithmetic progression, let them be m - n, m, and m + n. The sum of these roots is 3m, which equals the negative coefficient of the x^2 term, so 3m = 12 and m = 4. The product of the roots (m - n)m(m + n) equals the negative of the constant term, giving 4*(16 - n^2) = 28. Solving 64 - 4n^2 = 28 results in 4n^2 = 36, so n = 3.

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