Say we have an equation:$y=$ ${x^2} + 2x + 1$
Completing the square we get:
$\eqalign{
& y={x^2} + 2x + 1 \cr
& = {(x + 1)^2} – {(1)^2} + 1 \cr
& = {(x + 1)^2} \cr} $
The minimum point of this parabola is (-1,0)
What I would like to know is how/why does putting a quadratic equation in completing the square form give you the minimum point of a parabola? What is it about this form that corresponds to give you the minimum point? I hope i've made myself clear, if not please ask me to make myself clearer.
Thank you!
Best Answer
The expression $(x+a)^2+b$ attains its minimum at $x=-a$ because $(x+a)^2\ge 0$ for all real values of $x$ with equality if and only if $x=-a$.