Completing the square is a helpful math technique. It helps change quadratic equations into a special format. This format shows important details, especially in relation to graphs and parabolas. This method is especially useful for students in Year 11 Mathematics, where they learn to work with quadratic functions in different ways.

What Is Completing the Square?

A quadratic equation usually looks like this:

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

Here, $a$, $b$, and $c$ are numbers, and $a$ cannot be zero. Completing the square helps us rewrite the equation into something called vertex form:

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

In this equation, $(h, k)$ is the vertex of the parabola. The vertex is either the highest point or the lowest point on the graph, depending on whether $a$ is positive or negative.

How to Complete the Square

Here are the simple steps to change the quadratic equation by completing the square:

1. Factor Out the First Coefficient: If $a$ is not 1, take it out from the first two terms:

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

2. Find the Number to Complete the Square: Take half of the $x$ coefficient from inside the brackets, square it, and then add and subtract that number:

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

3. Write It as a Perfect Square: Now, rewrite the expression as a perfect square. Also, simplify any constant terms:

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

4. Combine the Constants: Combine the constant numbers to find $k$ in the vertex form:

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

Example

Let's look at this quadratic equation:

$$y = 2x^2 + 8x + 5$$

1. Factor out the 2 from the $x$ parts:

   $$y = 2(x^2 + 4x) + 5$$

2. Complete the square:

   Take half of 4 (which is 2), square it (which is 4), so:

   $$y = 2(x^2 + 4x + 4 - 4) + 5 = 2((x + 2)^2 - 4) + 5$$

3. Rewrite as a perfect square:

   $$y = 2(x + 2)^2 - 8 + 5$$

4. Combine the constants:

   $$y = 2(x + 2)^2 - 3$$

Now, the vertex $(h, k)$ is at $(-2, -3)$, which shows the lowest point on the parabola.

Why This Matters in Coordinate Geometry

The vertex form of the quadratic equation $(h, k)$ connects directly to how the parabola looks on a graph. You can easily see where the vertex is and if the parabola opens upward (if $a$ is positive) or downward (if $a$ is negative).

Also, you can change the graph in two ways:

• Vertical Shifts: Changing $k$ moves the graph up or down.
• Horizontal Shifts: Changing $h$ moves the graph left or right.

In short, completing the square helps with solving quadratic equations. It also helps students better understand parabolas in coordinate geometry. This foundation is very important for Year 11 math and prepares students for future studies in algebra and calculus.