Question

Completing the square??????

Answer 1

We want to write
`n^2+8n+59` in the form
`(n-a)^2+b`. Expand the second expression and set it equal to the first:

`n^2-2an+a^2+b=n^2+8n+59`

So we need to have

`\begin{cases}-2a=8\\a^2+b=59\end{cases}\implies a=-4,b=43`

Then

`n^2+8n+59=(n+4)^2+43=0\implies(n+4)^2=-43\implies n+4=\pm i√(43)`

`\implies n=-4\pm i√(43)`