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Given that the polynomial x^2 - 15x + t = 0 has only positive integer roots, find the average of all distinct possible values of t.

 Feb 21, 2022
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For the quadratic to have only positive roots,  \(\sqrt{b^2-4ac}\) must be an odd integer. 

 

This can only happen \(b^2 - 4ac\) is an odd square number. The only ways this can happen is if the discriminant is any 1 of these values: 1, 9, 25, 49, 81, 121, or 169. We can eliminate 225, because then the quadratic would have a root of 0, which isn't positive. 

 

For the values of the discriminant to happen, t must equal: 14, 26, 36, 44, 50, 54, or 56. From here, you can calculate the mean. 

 Feb 21, 2022

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