Question 1131744
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Let a, b and c be these roots.

Since they form an AP, we can write them as  m-d, m and m+d, where m is the middle term m=b and d is the common difference,  so

    a = m-d, c = m+d.


Then according to the Vieta's theorem, the sum of the roots is equal to the coefficient at  {{{x^2}}}  taken with the opposite sign:

    (m-d) + m + (m+d) = 9,   or   3m = 9,  which implies  m = 9/3 = 3.


The product of the roots, using the Vieta's theorem again, is equal to the constant term taken with the opposite sign:

    (3-d)*3*(3+d) = 15,   or  {{{3^2 - d^2}}} = 15/3 = 5,  which implies  {{{d^2}}} = 9 - 5 = 4;  hence,  d = +/-{{{sqrt(4)}}} = +/-2.


In this way, the AP is  <U>EITHER</U>   3-2 = 1, 3, 3+2 = 5  <U>OR</U>  5, 3, 1,  which makes no difference.


<U>Answer</U>.  The roots are  1, 3 and 5.
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Solved.