Completing the Square: A Simple Guide

Completing the square is an important way to change a quadratic equation from its standard form, which looks like this:
(y = ax^2 + bx + c)
to a format called vertex form:
(y = a(x - h)^2 + k).

This new format helps us see important features of the quadratic function.

Why Should You Use Completing the Square?

1. Finding the Vertex:

   • The vertex form shows you the vertex of the parabola right away.
   • The vertex is the highest or lowest point on the graph and is given by the coordinates ((h, k)).
   • For example, if the equation is:
     (y = 2(x - 3)^2 + 4),
     then the vertex is at the point ((3, 4)).

2. Understanding the Parabola:

   • The vertex form also makes it easy to see which way the parabola opens. If (a > 0), it opens up. If (a < 0), it opens down.
   • About half of students say they find it easier to notice how the graph changes (like flipping or stretching) when looking at vertex form.

3. Making Graphing Easier:

   • When you change the equation to vertex form, it's simpler to graph it.
   • You can start from the vertex and use the axis of symmetry to draw the parabola.
   • Many graphing tools and calculators use vertex form by default, which makes it even more useful.

4. Solving Quadratic Equations:

   • Completing the square helps you find solutions to the quadratic equation more easily.
   • You can also use this method to find the quadratic formula.

Steps to Complete the Square:

1. Begin with the standard form: (y = ax^2 + bx + c).
2. Factor out (a) from the first two terms.
3. Find ((\frac{b}{2a})^2) and add this to both sides of the equation.
4. Rewrite the left side as a squared binomial.
5. Rearrange everything into vertex form.

By learning how to complete the square, you’ll not only better understand quadratic functions but also gain essential math skills for the future. This makes it a key part of the Year 8 curriculum.