Multiple choice

If the roots of the equation $x^3-12x^2+39x-28=0$ are in A.P., then their common difference will be $\pm 3$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If roots are in AP, let them be a-d, a, a+d. Sum of roots = 3a = 12, so a = 4. Product of roots = (4-d)(4)(4+d) = 28. (16-d^2) = 7, d^2 = 9, d = +/- 3. The statement is true.

AI explanation

Assume the roots of the cubic equation are in arithmetic progression, expressed as a-d, a, and a+d. Using Vieta's formulas, the sum of these roots is 3a, which equals the coefficient 12, so a = 4. The product of the roots is (4-d)(4)(4+d) = 4(16-d^2) = 28, which simplifies to 16-d^2 = 7 and yields d^2 = 9. Taking the square root gives the common difference d as positive or negative 3, confirming the statement is true.

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