# The sum of two numbers is -29. The product of the same two numbers is 96. What are the two numbers?

Feb 14, 2018

The two numbers are $- 4$ and $- 24$.

#### Explanation:

You can translate the two statements from English to math:

$\stackrel{x + y}{\overbrace{\text{The sum of two numbers"" " stackrel(=) overbrace"is"" " stackrel(-28) overbrace"-28.}}}$

$\stackrel{x \cdot y}{\overbrace{\text{The product of the same two numbers"" " stackrel(=) overbrace"is"" " stackrel(96) overbrace"96.}}}$

Now we can create a system of equations:

$\left\{\begin{matrix}x + y = - 28 & q \quad \left(1\right) \\ x \cdot y = 96 & q \quad \left(2\right)\end{matrix}\right.$

Now, solve for $x$ in equation $\left(1\right)$:

$\textcolor{w h i t e}{\implies} x + y = - 28$

$\implies x = - 28 - y$

Plug this new $x$ value into equation $\left(2\right)$:

$\textcolor{w h i t e}{\implies} x \cdot y = 96$

$\implies \left(- 28 - y\right) \cdot y = 96$

$\textcolor{w h i t e}{\implies} - 28 y - {y}^{2} = 96$

$\textcolor{w h i t e}{\implies} - {y}^{2} - 28 y - 96 = 0$

$\textcolor{w h i t e}{\implies} {y}^{2} + 28 y + 96 = 0$

$\textcolor{w h i t e}{\implies} \left(y + 24\right) \left(y + 4\right) = 0$

$\textcolor{w h i t e}{\implies} y = - 4 , - 24$

Lastly, plug both of these $y$ values back into equation $\left(1\right)$:

For $y = - 4$:

$\textcolor{w h i t e}{\implies} x + y = - 28$

$\implies x - 4 = - 28$

$\textcolor{w h i t e}{\implies} x = - 24$

And for $y = - 24$:

$\implies x - 24 = - 28$

$\textcolor{w h i t e}{\implies} x = - 4$

Finally, we see that there are two solutions which are the same: $\left(- 4 , - 24\right)$ and $\left(- 24 , - 4\right)$.

This means that the two numbers are $- 4$ and $- 24$.