the product of two numbers is 18 and the difference is 3 what are the two numbers
MathBot Answer:
\(-3\), \(-6\) and \(6\), \(3\) are the numbers whose difference is \(3\), and whose product is \(18\).
The difference between the numbers is \(3\), and their product is \(18\). This translates to the two equations: \[\begin{aligned}n_{1} - n_{2} &= 3\\n_{1} n_{2} &= 18\\\end{aligned}\] Solving the first equation for \(n_{2}\) in terms of \(n_{1}\) produces the formula \(n_{2} = n_{1} - 3\).
Substituting \(n_{1} - 3\) for \(n_{2}\) in the second equation produces a quadratic equation that can be solved by the quadratic formula:\[\begin{gather*}n_{1} \cdot \left( n_{1} - 3 \right) = 18\\n_{1}^{2} - 3 n_{1} - 18 = 0\end{gather*}\] The solutions to the quadratic equation are \(n_{1} = -3\) and \(n_{1} = 6\).
Substituting \(n_{1} = -3\) into \(n_{1} - n_{2} = 3\), and solving for \(n_{2}\) yields \(n_{2} = -6\). Similarly, \(n_{1} = 6\) implies \(n_{2} = 3\).