find two numbers whose sum is 27 and product is 182
MathBot Answer:
The numbers are \(13\) and \(14\).
The sum of the numbers is \(27\), and their product is \(182\). This translates to the two equations: \[\begin{aligned}n_{1} + n_{2} &= 27\\n_{1} n_{2} &= 182\\\end{aligned}\] Solving the first equation for \(n_{2}\) in terms of \(n_{1}\) produces the formula \(n_{2} = 27 - n_{1}\).
Substituting \(27 - n_{1}\) for \(n_{2}\) in the second equation produces a quadratic equation that can be solved by the quadratic formula:\[\begin{gather*}n_{1} \cdot \left( 27 - n_{1} \right) = 182\\n_{1}^{2} - 27 n_{1} + 182 = 0\end{gather*}\] The solutions to the quadratic equation are \(n_{1} = 13\) and \(n_{1} = 14\).
Substituting \(n_{1} = 13\) into \(n_{1} + n_{2} = 27\), and solving for \(n_{2}\) yields \(n_{2} = 14\). Similarly, \(n_{1} = 14\) implies \(n_{2} = 13\).