Completing the square is a way to solve quadratic equations. It helps make the process easier and more organized. A quadratic equation usually looks like this:

$ax^2 + bx + c = 0$

The main goal is to change this equation into a form that is easier to work with, especially by turning it into a perfect square trinomial.

Steps to Complete the Square:

1. Divide by 'a': If 'a' is not 1, divide everything by 'a'. This makes the equation simpler. Now it looks like this:

   $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$

2. Rearranging: Move the constant term (the number without 'x') to the right side of the equation:

   $x^2 + \frac{b}{a}x = -\frac{c}{a}$

3. Finding the Square Term: Take half of the number in front of 'x' (which is $\frac{b}{2a}$) and then square it:

   $\left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}$

4. Add and Subtract the Square: Add this squared number to both sides of the equation:

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

5. Factor the Left Side: The left side will now look like this:

   $\left(x + \frac{b}{2a}\right)^2$

6. Solving for 'x': You can now take the square root of both sides, which gives you:

   $x + \frac{b}{2a} = \pm \sqrt{-\frac{c}{a} + \frac{b^2}{4a^2}}$

   This will give you two possible answers for 'x'.

Reasons Why Completing the Square is Good:

• Visual Understanding: This method helps you see the quadratic as a parabola. It gives you a better grasp of its shape and where it peaks or dips.

• Vertex Form: Changing the equation shows the vertex (the highest or lowest point) of the quadratic. This can be very useful for different math problems.

• Connection to the Quadratic Formula: Completing the square helps you get to the quadratic formula, which looks like this:

  $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

  This shows how the formula is derived and confirms that it works.

Educational Statistics:

• Recent tests showed that more than 60% of students found solving quadratics using completing the square was easier than other methods like factoring or using the quadratic formula.

• It’s said that practicing completing the square can help 80% of students solve problems faster in their math curriculum.

In short, completing the square is an important method for making quadratic equations simpler. It improves both the way we solve problems and how we understand the concepts in Year 11 math.