Question 389639
The easiest way is probably to guess and check, since it is likely that both numbers will be integers. Guessing and checking, the numbers are 9 and 8.


Or, we can use {{{p + q = 17}}} and {{{p^2 + q^2}}} = 145. Squaring the first equation,


{{{(p + q)^2 = p^2 + 2pq + q^2 = 289}}} --> {{{2pq = 144}}} --> {{{pq = 72}}}.


If we assume that p and q are roots of a polynomial {{{x^2 + bx + c}}}, use Vieta's formula:


The sum of the roots is -b --> {{{p + q = 17 = -b}}}
The product of the roots is c --> {{{pq = 72 = c}}}


This implies b = -17 and c = 72, so p and q are roots of the polynomial {{{x^2 - 17x + 72}}}. The polynomial can be factored as {{{(x - 8)(x - 9)}}} so p,q = (8,9).