Completing the Square (AQA GCSE Maths): Revision Notes

Completing the square

What is completing the square?

Completing the square is a powerful algebraic technique used to solve quadratic equations and analyse their graphs. While it might seem challenging at first, once you understand the systematic approach, it becomes a valuable tool for working with quadratic expressions.

The method involves rewriting a quadratic expression in the form $(x + m)^2 + n$, which reveals important information about the quadratic's behaviour and makes certain calculations much easier.

The basic method (when a = 1)

When working with quadratics where the coefficient of $x^2$ is 1, you can follow this systematic four-step process.

The process works by transforming expressions like $x^2 + 8x + 5$ into the completed square form. Here's how each step works:

Step 1: Rearrange into standard format Make sure your quadratic is written as $ax^2 + bx + c$. For basic cases, we work with expressions where $a = 1$.

Step 2: Write the initial bracket Create a bracket $(x + \frac{b}{2})^2$ where you divide the coefficient of x by 2. This is the foundation of your completed square.

Step 3: Multiply out the brackets and compare Expand your initial bracket and compare it to the original expression. This shows you what adjustments need to be made.

Step 4: Add the adjusting number Calculate what number you need to add or subtract to make the expanded bracket match the original expression exactly.

Using the completed square to solve equations

Once you have the completed square form, solving quadratic equations becomes more straightforward. The process involves three additional steps:

1. Rearrange the equation so the completed square equals zero
2. Square root both sides (remembering to include both positive and negative solutions)
3. Solve for x by isolating the variable

For example, if you have $(x + 4)^2 - 11 = 0$, you can rearrange to get $(x + 4)^2 = 11$, then take the square root to find $x + 4 = \pm\sqrt{11}$, giving you $x = -4 \pm \sqrt{11}$.

When the coefficient of x² is not 1

When dealing with quadratics where $a \neq 1$, you need to factor out the coefficient of $x^2$ first before applying the completing the square method.

The process becomes slightly more complex because you need to work with fractions. This requires careful attention to fraction arithmetic, but the fundamental principle remains the same.

Using the completed square to sketch graphs

The completed square form provides valuable information for sketching quadratic graphs. The form $a(x + m)^2 + n$ reveals two crucial pieces of information:

Finding the turning point: The minimum (or maximum) point occurs when the bracket equals zero, which happens when $x = -m$. At this point, the y-value equals $n$.

Determining if the graph crosses the x-axis: If the adjusting number ($n$) is positive for a positive quadratic, the graph never crosses the x-axis because the minimum value is above zero.

Important shortcuts and tips

For those who find the full method challenging, remember that the adjusting number always follows the pattern $c - \left(\frac{b}{2}\right)^2$. This can help you check your work or speed up calculations.