two numbers multiply to get -30 and add together to get -1 what are the two numbers
MathBot Answer:
The numbers are \(-6\) and \(5\).
The sum of the numbers is \(-1\), and their product is \(-30\). This translates to the two equations: \[\begin{aligned}n_{1} + n_{2} &= -1\\n_{1} n_{2} &= -30\\\end{aligned}\] Solving the first equation for \(n_{2}\) in terms of \(n_{1}\) produces the formula \(n_{2} = - n_{1} - 1\).
Substituting \(- n_{1} - 1\) for \(n_{2}\) in the second equation produces a quadratic equation that can be solved by the quadratic formula:\[\begin{gather*}n_{1} \cdot \left( - n_{1} - 1 \right) = -30\\n_{1}^{2} + n_{1} - 30 = 0\end{gather*}\] The solutions to the quadratic equation are \(n_{1} = -6\) and \(n_{1} = 5\).
Substituting \(n_{1} = -6\) into \(n_{1} + n_{2} = -1\), and solving for \(n_{2}\) yields \(n_{2} = 5\). Similarly, \(n_{1} = 5\) implies \(n_{2} = -6\).