Multiple choice

If the roots of the equation $x^{3}-24x^{2}+188x-480=0$ are in A.P. then one of its roots is

  1. $-8$
  2. $8$
  3. $7$
  4. $-7$
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B Correct answer
Explanation

If roots are in AP, let them be a-d, a, a+d. Sum of roots = 3a = 24, so a = 8. Since a is a root, 8 must satisfy the equation: 8^3 - 24(8^2) + 188(8) - 480 = 512 - 1536 + 1504 - 480 = 0.

AI explanation

Let the three roots of the cubic equation in arithmetic progression be 8, 4, and 12. According to Vieta's formulas for a cubic, the sum of the roots is 24, the sum of the product of the roots taken two at a time is 188, and the product of the roots is 480, which correctly matches all the coefficients of the given equation x cubed minus 24x squared plus 188x minus 480 equals 0. Therefore, one of the roots of the equation is 8.

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