Suppose that $k_1k_2=\alpha$ and that $k_1+k_2=\gamma$. Are $k_1$ and $k_2$ unique? Let $k_1, k_2, \alpha, \gamma \in \mathbb{Z}$.
Suppose that $k_1k_2=\alpha$ and that $k_1+k_2=\gamma$.
Are $k_1$ and $k_2$ unique? That is, can one find another pair of integers whose sum is also $\alpha$ and whose product is also $\gamma$?
If so, how can one prove this?
 A: Suppose you have
$$k_1k_2=\alpha \tag{1}\label{eq1A}$$
$$k_1+k_2=\gamma \tag{2}\label{eq2A}$$
WLOG, assume $k_1 \le k_2$. Assume there's a $k_3$ and $k_4$ where $k_3 \le k_4$, $k_3 \neq k_1$ such that
$$k_3+k_4=\gamma \tag{3}\label{eq3A}$$
There's some $d \neq 0$ where
$$k_3 = k_1 + d \tag{4}\label{eq4A}$$
Then you get
$$\begin{equation}\begin{aligned}
k_4 & = \gamma - k_3 \\
k_4 & = k_1 + k_2 - (k_1 + d) \\
k_4 & = k_2 - d
\end{aligned}\end{equation}\tag{5}\label{eq5A}$$
You also have
$$k_3k_4 = (k_1 + d)(k_2 - d) = k_1k_2 - d^2 = \alpha - d^2 \tag{6}\label{eq6A}$$
This shows that, apart from order, the values of $k_1$ and $k_2$ are unique.
A: Yes they are unique up to a swapping operation.
By means of the identity
$$ k_1 k_2 = [ (k_1+k_2)/2]^2-[ (k_1-k_2)/2]^2 $$
which is the same as
$$\alpha =  \gamma^2/4-[ (k_1-k_2)/2]^2 $$
we see that $(k_1,k_2)$ can be interchanged for same $\alpha,\gamma.$
If the sign of $\alpha $ is changed, then either one of $k_1$ and $k_2$ can undergo a sign change.
For example for product $+25$ if $(k_1,k_2)=(1,25)$ or $ (25,1)$ the sum is unchanged at $26$ by associative law.
Similarly for product $-25$ if $(k_1,k_2)=(-1,25)$ or $ (-25,1)$ the sum is unchanged at $-12.$
It can be also shown geometrically in Mohr's Circle of stress/strain.
