Completing the Square and Finding Max/Min Points

Ever wondered how to find the exact turning point of a parabola without drawing the entire graph? Completing the square transforms any quadratic into the form $(x + p)^2 + q$, where the coordinates $(-p, q)$ give you the vertex directly.

The formula is straightforward: $x^2 + bx + c = (x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c$. Take half of the coefficient of $x$, square it, add it, then subtract it to keep the expression equal.

Let's work through $x^2 + 9x + 4 = 0$. Here, $b = 9$, so we get $(x + \frac{9}{2})^2 - (\frac{9}{2})^2 + 4 = (x + \frac{9}{2})^2 - \frac{65}{4}$. The minimum point is at $(-\frac{9}{2}, -\frac{65}{4})$ because the coefficient of $x^2$ is positive.

When the leading coefficient isn't 1, like in $4x^2 + 4x + 2$, factor it out first:$4$x^2 + x + \frac{1}{2}$$. Complete the square inside the brackets, then multiply back through to get$4$x + \frac{1}{2}$^2 + 1$. The minimum point is at$$-\frac{1}{2}, 1$$.

Quick Tip: If the coefficient of $x^2$ is positive, you've found a minimum point. If it's negative, it's a maximum point!