Where does the plus-minus come from in the quadratic formula? In the formula
$$x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
for solving quadratic equations, where does the $\pm$ come from?
The square root already results in both a positive and negative term, is the $\pm$ not therefore extraneous?

What about completing squares?
\begin{align*}
ax^{2} + bx + c & = a\left(x^{2} + \frac{bx}{a}\right) + c\\\\
& = a\left(x^{2} + \frac{bx}{a} + \frac{b^{2}}{4a^{2}}\right) + c - \frac{b^{2}}{4a}\\\\
& = a\left(x + \frac{b}{2a}\right)^{2} + \frac{4ac - b^{2}}{4a}
\end{align*}
Hence we deduce that
\begin{align*}
ax^{2} + bx + c = 0 & \Longleftrightarrow a\left(x + \frac{b}{2a}\right)^{2} + \frac{4ac - b^{2}}{4a} = 0\\\\
& \Longleftrightarrow \left(x + \frac{b}{2a}\right)^{2} = \frac{b^{2} - 4ac}{4a^{2}}\\\\\
& \Longleftrightarrow x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} 
\end{align*}
and we are done.
Hopefully this helps!
