Completing the Square

Definition A square term is an expression of the form $(x + p)^2$ with $p$ a real constant. For every real $x$,

A quadratic $x^2 + bx + c$ is a perfect square exactly when $c = \left(\frac{b}{2}\right)^2$, and then it equals $\left(x + \frac{b}{2}\right)^2$.

Definition For every real $x$ and every real constant $b$,

Completing the square means using this identity to rewrite a quadratic expression as a square term, or a constant multiple of one, plus a constant.

Definition Adding a number $k$ and subtracting the same $k$ changes no value, since $E = E + k - k$ for every expression $E$. In completing the square the adjustment is $k = \left(\frac{b}{2}\right)^2$. It is added to build the perfect square and subtracted at once to keep the expression equal to what it was.

Definition The leading coefficient of $ax^2 + bx + c$ is $a$, assumed nonzero. To complete the square, factor $a$ out of the two terms that contain $x$,

and then complete the square inside the parentheses. Anything added inside the parentheses is multiplied by $a$, so the outside compensation must be $a$ times the inside adjustment.

Definition Completing the square turns $ax^2 + bx + c$ with $a \ne 0$ into the vertex form

This is an identity, true for every real $x$, so the two forms are equivalent everywhere. The graph of $y = ax^2 + bx + c$ is a parabola with vertex $(h, k)$.

Definition For every real $x$ we have $(x - h)^2 \ge 0$, with equality only at $x = h$. If $a > 0$, then $a(x - h)^2 \ge 0$ as well, so

with equality exactly at $x = h$. The minimum value of the quadratic is $k$, attained only at $x = h$, and there is no maximum. If $a < 0$ the inequality reverses, $k$ is the maximum value, and there is no minimum.