If the roots of the equation x3 - 12x2 + 39x - 28 = 0 are in A.P., the...
Given:
The equation is x^3 - 12x^2 + 39x - 28 = 0.
To find:
The common difference of the roots if they are in arithmetic progression (A.P.).
Solution:
Let the roots of the equation be a - d, a, and a + d, where a is the middle term and d is the common difference.
Sum of the roots:
According to Vieta's formulas, the sum of the roots of a cubic equation is given by the negative coefficient of the quadratic term divided by the coefficient of the cubic term.
In this case, the sum of the roots is -(-12)/1 = 12.
Using the sum of roots:
The sum of the roots of the equation is given by (a - d) + a + (a + d) = 12.
Simplifying, we get 3a = 12, which gives a = 4.
Product of the roots:
According to Vieta's formulas, the product of the roots of a cubic equation is given by the constant term divided by the coefficient of the cubic term.
In this case, the product of the roots is -(constant term)/(coefficient of cubic term) = -(-28)/1 = 28.
Using the product of roots:
The product of the roots of the equation is given by (a - d)(a)(a + d) = 28.
Expanding, we get a^3 - d^2a = 28.
Using the values of a:
Substituting a = 4 in the equation a^3 - d^2a = 28, we get 64 - 16d^2 = 28.
Simplifying, we get 16d^2 = 36, which gives d^2 = 9.
Taking the square root of both sides, we get d = ±3.
Conclusion:
The common difference of the roots is ±3. However, since the options only include positive integers, the common difference is 3.
Therefore, the correct answer is option C: 3.