[tex]\begin{cases} \stackrel{\textit{sum is twelve}}{a + b = 12}\\ \stackrel{\textit{product is 32}}{ab = 32}\\[-0.5em] \hrulefill\\ a + b = 12\\ b = 12 - a \end{cases}\qquad \implies \stackrel{\textit{substituting that in our 2nd equation}~\hfill }{a(12-a) = 32\implies 12a-a^2=32} \\\\\\ 0=a^2-12a+32\implies 0=(a-8)(a-4)\implies a= \begin{cases} 8\\ 4 \end{cases} \\\\\\ \stackrel{\textit{we know that}}{b = 12 - a}\implies b = \begin{cases} 12-8\to &4\\ 12-4\to &8 \end{cases}[/tex]
