Math Problem Solver

The numbers are \(6\) and \(12\).

The sum of the numbers is \(18\), and their product is \(72\). This translates to the two equations: \[\begin{aligned}n_{1} + n_{2} &= 18\\n_{1} n_{2} &= 72\\\end{aligned}\] Solving the first equation for \(n_{2}\) in terms of \(n_{1}\) produces the formula \(n_{2} = 18 - n_{1}\).
Substituting \(18 - n_{1}\) for \(n_{2}\) in the second equation produces a quadratic equation that can be solved by the quadratic formula:\[\begin{gather*}n_{1} \cdot \left( 18 - n_{1} \right) = 72\\n_{1}^{2} - 18 n_{1} + 72 = 0\end{gather*}\] The solutions to the quadratic equation are \(n_{1} = 6\) and \(n_{1} = 12\).
Substituting \(n_{1} = 6\) into \(n_{1} + n_{2} = 18\), and solving for \(n_{2}\) yields \(n_{2} = 12\). Similarly, \(n_{1} = 12\) implies \(n_{2} = 6\).