Completing the square is a helpful way to change a quadratic equation into a form that makes it easy to find the vertex of a parabola.

A standard quadratic equation looks like this:

$$y = ax^2 + bx + c$$

In this equation, $a$, $b$, and $c$ are numbers called constants. The vertex of a parabola is the highest or lowest point, and it is represented by the coordinates $(h, k)$.

Steps to Complete the Square

1. Factor Out the Coefficient of $x^2$: If $a$ is not equal to 1, we begin by taking $a$ out of the first two terms:

   $$y = a(x^2 + \frac{b}{a}x) + c$$

2. Compute the Square: To complete the square, we take half of the number in front of $x$, square it, and then add and subtract this value inside the parentheses:

   $$y = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c$$

   Now, this changes the equation to:

   $$y = a\left(\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c$$

3. Simplify the Equation: Next, we distribute $a$ and combine the constant terms to get:

   $$y = a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b}{2a}\right)^2 + c$$

   Now we can write it in vertex form:

   $$y = a\left(x - h\right)^2 + k$$

   where:

   $$h = -\frac{b}{2a} \quad \text{and} \quad k = c - \frac{b^2}{4a}$$

Finding the Vertex

1. Identify $h$ and $k$: From our vertex form, we can find:

   • The $x$-coordinate of the vertex, $h$, can be figured out using this formula:

     $$h = -\frac{b}{2a}$$

   • To find the $y$-coordinate, $k$, we substitute $h$ back into the original equation:

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

2. Vertex Location: So, the vertex of the parabola can be found at the point:

   $$(h, k) = \left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)$$

Conclusion

Completing the square makes it easier to find important points of a parabola. It shows how the numbers in the equation relate to the shape of the graph. This method is commonly used in algebra, and knowing how to do it can improve your ability to work with quadratic functions. Once you master these steps, you'll feel confident finding the vertex and understanding how parabolas behave in different situations.