Question about solution to Laplacian in Spherical Polars

[Sum Guy]

I was following this derivation of the solution to the Laplacian in spherical polars. I was wondering where the two equations $\lambda_{1} + \lambda_{2} = -1$ and $\lambda_{1}\lambda_{2} = -\lambda$ come from? Thanks.

 

[ShayanJ]

We want to solve the equation $\frac{d^2S}{dt^2}+\frac{dS}{dt}-\lambda S=0$. So let's try a solution of the form $S=e^{bt}$ (and find b), which gives $b^2+b-\lambda=0$. But for any quadratic equation $px^2+qx+r=0$, we know that the two roots satisfy $x_1+x_2=-\frac{q}{p}$ and $x_1x_2=\frac{r}{p}$.