what 2 numbers add together to make -6 and multiply to make -8
MathBot Answer:
The numbers are \(- \sqrt{17} - 3\) and \(-3 + \sqrt{17}\).
The sum of the numbers is \(-6\), and their product is \(-8\). This translates to the two equations: \[\begin{aligned}n_{1} + n_{2} &= -6\\n_{1} n_{2} &= -8\\\end{aligned}\] Solving the first equation for \(n_{2}\) in terms of \(n_{1}\) produces the formula \(n_{2} = - n_{1} - 6\).
Substituting \(- n_{1} - 6\) for \(n_{2}\) in the second equation produces a quadratic equation that can be solved by the quadratic formula:\[\begin{gather*}n_{1} \cdot \left( - n_{1} - 6 \right) = -8\\n_{1}^{2} + 6 n_{1} - 8 = 0\end{gather*}\] The solutions to the quadratic equation are \(n_{1} = - \sqrt{17} - 3\) and \(n_{1} = -3 + \sqrt{17}\).
Substituting \(n_{1} = - \sqrt{17} - 3\) into \(n_{1} + n_{2} = -6\), and solving for \(n_{2}\) yields \(n_{2} = -3 + \sqrt{17}\). Similarly, \(n_{1} = -3 + \sqrt{17}\) implies \(n_{2} = - \sqrt{17} - 3\).