Completing the square is a helpful way to find the peak point, called the vertex, of a quadratic function. A quadratic function usually looks like this:

$y = ax^2 + bx + c$

Here’s How to Complete the Square:

1. Start with the equation: Write it in a way that puts the quadratic and linear parts together. $y = ax^2 + bx + c$

2. Factor out the leading number (if it’s not 1): $y = a(x^2 + \frac{b}{a}x) + c$

3. Complete the square: Take half of the number in front of $x$, square it, and then change the equation. $y = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c$ This changes to: $y = a\left(\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c$

4. Simplify: Rewrite it like this: $y = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)$

Finding the Vertex:

Now you can easily find the vertex $(h, k)$ of the parabola:

• $h = -\frac{b}{2a}$
• $k = c - \frac{b^2}{4a}$

Example: Let’s look at this function, $y = 2x^2 + 8x + 5$:

1. First, factor out the 2: $y = 2(x^2 + 4x) + 5$.
2. Now complete the square: $y = 2((x + 2)^2 - 4) + 5$.
3. Simplifying gives us $y = 2(x + 2)^2 - 3$.
4. So, the vertex is at the point $(-2, -3)$.

Completing the square is not just about solving equations. It also helps you see where the vertex is on a graph!