Solve the equation x3 – 13x2 + 15x + 189 = 0, Given that one of the roots exceeds theother by 2.

Sure, let's solve this step by step.

Step 1: Given that one root exceeds the other by 2, let's assume the roots to be (a), (a+2), and (b).

Step 2: According to Vieta's formulas, the sum of the roots taken one at a time is equal to the negation of the coefficient of the second term divided by the coefficient of the first term. So, a + (a+2) + b = 13.

Step 3: Simplifying the above equation, we get 2a + b + 2 = 13, which simplifies further to 2a + b = 11.

Step 4: Also, according to Vieta's formulas, the sum of the roots taken two at a time is equal to the coefficient of the third term divided by the coefficient of the first term. So, a*(a+2) + ab + b(a+2) = 15.

Step 5: Simplifying the above equation, we get a^2 + 2a + ab + 2b = 15.

Step 6: Substituting 2a + b = 11 in the above equation, we get 11a + 22 = 15, which simplifies to 11a = -7, and further simplifies to a = -7/11.

Step 7: Substituting a = -7/11 in 2a + b = 11, we get b = 11 - 2*(-7/11) = 15/11.

Step 8: So, the roots of the equation are (-7/11), (-7/11 + 2) = 15/11, and 15/11.