Math Problem Solver

The numbers are \(19\) and \(19\).

The sum of the numbers is \(38\), and their product is \(361\). This translates to the two equations: \[\begin{aligned}n_{1} + n_{2} &= 38\\n_{1} n_{2} &= 361\\\end{aligned}\] Solving the first equation for \(n_{2}\) in terms of \(n_{1}\) produces the formula \(n_{2} = 38 - n_{1}\).
Substituting \(38 - n_{1}\) for \(n_{2}\) in the second equation produces a quadratic equation that can be solved by the quadratic formula:\[\begin{gather*}n_{1} \cdot \left( 38 - n_{1} \right) = 361\\n_{1}^{2} - 38 n_{1} + 361 = 0\end{gather*}\] The solution to the quadratic equation is \(n_{1} = 19\).
Substituting \(n_{1} = 19\) into \(n_{1} + n_{2} = 38\), and solving for \(n_{2}\) yields \(n_{2} = 19\).