Completing the Square Revision Notes for Edexcel GCSE Maths

Completing the Square

Completing the square is a powerful algebraic technique used to solve quadratic equations. While it might seem tricky at first, once you understand the steps, it becomes a valuable tool for working with quadratics that don't factorise easily.

What is completing the square?

When we complete the square, we transform a quadratic expression from its standard form ( $ax^2 + bx + c$ ) into a perfect square format plus or minus a constant. This makes it much easier to solve equations and understand the properties of quadratic functions.

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The key insight is that we're essentially converting a quadratic expression into the form $(x + p)^2 + q$ , where $p$ and $q$ are constants. This reveals important information about the quadratic, such as its vertex and makes solving much more straightforward.

The four-step method

The process follows these four clear steps:

Step 1: Rearrange into standard format Make sure your quadratic is written as $ax^2 + bx + c$ . For most problems, you'll be working with $a = 1$ , which makes the process more straightforward.

Step 2: Write the initial bracket Take the coefficient of x (which is $b$ ), divide it by 2, and write $(x + \frac{b}{2})^2$ . This forms the foundation of your completed square.

Step 3: Multiply out and compare Expand your bracket from step 2 and compare it to your original expression. You'll notice that the constant term doesn't match - this is where the adjusting number comes in.

Step 4: Add the adjusting number Subtract or add the appropriate number to make your expression equal to the original. This gives you the completed square form.

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The most common mistake is forgetting to add the adjusting number in step 4. Always remember to compare your expanded bracket with the original expression and adjust accordingly.

Worked example

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Worked Example: Completing the Square for $x^2 + 8x + 5$

Let's see how this works with $x^2 + 8x + 5$ :

Step 1: The expression is already in standard form: $x^2 + 8x + 5$

Step 2: The coefficient of $x$ is 8, so we divide by 2 to get 4. Our initial bracket is $(x + 4)^2$

Step 3: When we expand $(x + 4)^2$ , we get $x^2 + 8x + 16$ . Comparing this to our original $x^2 + 8x + 5$ , we can see we have $+16$ instead of $+5$ .

Step 4: We need to subtract 11 to match the original, giving us $(x + 4)^2 - 11$ .

Therefore: $x^2 + 8x + 5 = (x + 4)^2 - 11$

Solving equations using completed square

Once you've completed the square, solving the equation becomes much more manageable.

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Worked Example: Solving $x^2 + 8x + 5 = 0$

Here's how to solve $x^2 + 8x + 5 = 0$ :

Step 1: Start with your completed square form: $(x + 4)^2 - 11 = 0$

Step 2: Move the constant to the other side: $(x + 4)^2 = 11$

Step 3: Take the square root of both sides: $x + 4 = \pm\sqrt{11}$

Step 4: Solve for $x$ : $x = -4 \pm \sqrt{11}$

This gives you two solutions: $x = -4 + \sqrt{11}$ and $x = -4 - \sqrt{11}$

Quick formula tip

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Time-saving shortcut: When the coefficient of $x^2$ is 1, there's a handy shortcut. The adjusting number you need to subtract is always $c - \left(\frac{b}{2}\right)^2$ , where $b$ is the coefficient of $x$ and $c$ is the constant term. This can save you time once you're comfortable with the method.

Why use completing the square?

Completing the square is particularly useful when:

The quadratic doesn't factorise easily
You need exact answers in surd form
You're working with quadratic functions and need to find the vertex
You're deriving the quadratic formula

The method might seem complex initially, but with practice, it becomes a reliable technique for handling challenging quadratic problems.

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Key Points to Remember:

Always start by ensuring your quadratic is in standard form ( $ax^2 + bx + c$ )
The key to the initial bracket is halving the coefficient of $x$
Don't forget to adjust by adding or subtracting the appropriate constant
When solving equations, remember that square rooting gives you $\pm$
Leave your final answers in surd form unless asked to give decimals

Completing the Square (Edexcel GCSE Maths): Revision Notes