📂Lemmas

Finding the Extremum of a Quadratic Function Quickly

Formula

The vertex of the quadratic function $f(x)=c(x-a)(x-b)$ is $\frac { a+b }{ 2 }$ (where $c\neq 0$)

For factorable quadratic functions, the vertex can be found without bothering with various calculations. It seems obvious, but knowing or not knowing this fact can make a difference in whether or not you can reduce one step in the calculation process.

Derivation

$$\begin{align*} & f(x) = c(x-a)(x-b) = c x^2 -c(a+b)x+cab \\ \implies& f '(x)=2cx-c(a+b) \\ \implies& 2cx-c(a+b)=0 \\ \implies& x=\frac { c(a+b) }{ 2c } \\ \implies& x=\frac { a+b }{ 2 } \end{align*}$$

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