Multiple choice

The difference between two roots of the equation ${ x }^{ 3 }-13{ x }^{ 2 }+15x+189=0$ is 2, then the roots of the equation are

  1. -3,7,9

  2. -3,-7,-9

  3. 3,-5,7

  4. -3,-7,9
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A Correct answer
Explanation

Let roots be a-1, a+1, b. Sum = 2a+b = 13. Product = (a^2-1)b = -189. Testing values, if roots are -3, 7, 9, sum = 13, product = -189. Difference between 9 and 7 is 2.

AI explanation

Let the two roots that differ by 2 be alpha and beta, meaning alpha - beta = 2. Using Vieta's formulas for x^3 - 13x^2 + 15x + 189 = 0, the sum of all three roots is 13, the sum of their pairwise products is 15, and their overall product is -189. Let the third root be gamma, so alpha + beta + gamma = 13 and alpha*beta*gamma = -189. Testing the options provided, the set -3, 7, and 9 satisfies all conditions: their sum is 13, their pairwise products sum to -21 + 63 - 27 = 15, their product is -189, and the difference between 9 and 7 is exactly 2. Therefore, the roots are -3, 7, and 9.

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