Math Problem Solver

The numbers are \(12\) and \(17\).

The sum of the numbers is \(29\), and their product is \(204\). This translates to the two equations: \[\begin{aligned}n_{1} + n_{2} &= 29\\n_{1} n_{2} &= 204\\\end{aligned}\] Solving the first equation for \(n_{2}\) in terms of \(n_{1}\) produces the formula \(n_{2} = 29 - n_{1}\).
Substituting \(29 - n_{1}\) for \(n_{2}\) in the second equation produces a quadratic equation that can be solved by the quadratic formula:\[\begin{gather*}n_{1} \cdot \left( 29 - n_{1} \right) = 204\\n_{1}^{2} - 29 n_{1} + 204 = 0\end{gather*}\] The solutions to the quadratic equation are \(n_{1} = 12\) and \(n_{1} = 17\).
Substituting \(n_{1} = 12\) into \(n_{1} + n_{2} = 29\), and solving for \(n_{2}\) yields \(n_{2} = 17\). Similarly, \(n_{1} = 17\) implies \(n_{2} = 12\).