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Lesson 5

Completing the Square to Solve Quadratic Equations

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Welcome to our fifth lesson! In our last session, we mastered solving quadratic equations by factorising. You learned that this powerful method relies on the Zero Product Property, but we also noted its main limitation: it only works if the quadratic expression can be factorised into simple terms.

Today, we'll learn a universal technique that can solve any quadratic equation: completing the square. This method is not just a solving algorithm; it's a way to restructure a quadratic expression to reveal its underlying properties. This is something that has direct parallels in programming and game development, where refactoring code or restructuring data can make a complex problem much simpler to solve. Mastering this will not only help you solve equations but will also give you the key to understanding the graphs of quadratic functions, which we'll explore later in the course.

Part 1: The Intuition Behind "Completing the Square"

Before we dive into the algebraic steps, let's build an intuition for what the name "completing the square" actually means. It has a very literal, geometric origin.

Consider an expression like x^2 + 10x. We can represent x^2 as a square with side length x, and 10x as a rectangle with sides 10 and x. To make a larger square, we can split the 10x rectangle into two smaller rectangles of 5x each and place them on the sides of the x^2 square.

This diagram shows how the algebraic terms x², 2bx, and b² can be represented as geometric shapes. The expression x² + 2bx (represented by the first three shapes) is an incomplete square. By adding a small square of area b², we "complete" the larger square, which has an area of (x+b)².

As you can see in the diagram, arranging x^2 and the two bx pieces leaves a small corner piece missing. The area of this missing piece is what "completes" the larger square. If the side of the original square is x and the rectangles we add on have a width of b, the missing piece is a square of b by b, or b^2.

This is the key insight: the expression x^2 + 2bx is almost a perfect square. To make it a perfect square, (x+b)^2, we need to add the term b^2.

Algebraically, we know that (x+a)^2 = x^2 + 2ax + a^2. If we have an expression like x^2 + bx, we can match the terms:

  • The x^2 term matches.
  • The x term tells us that b must equal 2a. This means a = b/2.
  • Therefore, the constant term we need to add to "complete the square" must be a^2, which is (b/2)^2.

The following reading explains this logic and gives us a simple rule.

Solving quadratics by completing the square (article)

This article from Khan Academy explains the logic of how to find the constant term needed to create a perfect square trinomial.

Read the section titled "How to complete the square". Focus on the connection between the expansion of (x+a)^2 and the formula for finding the missing term: (b/2)^2.

Part 2: The Method for Solving Equations

Now that you understand the "why," let's apply this to solve an equation. As per your preference, try this one first. Notice that you can't easily factorise the left-hand side.

Solve for x: x^2 - 6x - 16 = 0

Take a moment to think. How could you transform the left side into a perfect square?

...

Let's walk through the solution using the completing the square method. The image below shows the full process.

This image demonstrates the step-by-step process of solving the quadratic equation x² - 6x - 16 = 0 using the completing the square method. It also shows the corresponding graph of the function, highlighting the solutions as the x-intercepts.

Here is a breakdown of the steps shown in the image:

  1. Rearrange: Isolate the terms with x on one side. We move the constant -16 to the right side by adding 16 to both sides.
    x^2 - 6x = 16

  2. Complete the Square: Find the term to add. Here, b = -6.
    The term we need is (b/2)^2 = (-6/2)^2 = (-3)^2 = 9.
    Add this value to both sides of the equation to keep it balanced.
    x^2 - 6x + 9 = 16 + 9

  3. Factor and Solve: The left side is now a perfect square. Factor it and simplify the right side.
    (x - 3)^2 = 25
    Now we can solve by taking the square root of both sides. Crucially, remember the positive and negative roots.
    x - 3 = ±sqrt(25)
    x - 3 = ±5

    This gives us two separate linear equations to solve:

    • x - 3 = 5 --> x = 8
    • x - 3 = -5 --> x = -2

The solutions are x = 8 and x = -2.

The following video from Maths Genie demonstrates this exact process of using the completed square form to solve an equation.

Completing The Square

This video provides a clear, step-by-step example of solving a quadratic equation by completing the square.

Watch the final segment of the video, from the beginning of the question, which shows how to rearrange the equation, complete the square, and solve for x, leaving the answer in surd form (which is common in exams when the solution isn't a whole number).

Part 3: Dealing with More Complex Quadratics

The method is straightforward when the coefficient of x^2 (the 'a' term) is 1. But what happens when a is not 1? For example, in an equation like 3x^2 - 12x - 18 = 0.

The rule for completing the square, (b/2)^2, only works when a=1. So, our first step must always be to make it so. We do this by factoring out the coefficient a.

Let's try solving 3x^2 - 12x - 18 = 0.

  1. Rearrange: Move the constant term to the right side.
    3x^2 - 12x = 18

  2. Factor out 'a': Divide both sides by a (which is 3) to make the x^2 coefficient equal to 1.
    x^2 - 4x = 6

  3. Complete the Square: Now we can proceed as before. Here, b = -4.
    The term to add is (b/2)^2 = (-4/2)^2 = (-2)^2 = 4.
    Add 4 to both sides:
    x^2 - 4x + 4 = 6 + 4

  4. Factor and Solve:
    (x - 2)^2 = 10
    x - 2 = ±sqrt(10)
    x = 2 ±sqrt(10)

The solutions are x = 2 + sqrt(10) and x = 2 - sqrt(10).

This process of first isolating the variable terms and then ensuring the x^2 coefficient is 1 is a fixed algorithm you can always follow.

Solving quadratics by completing the square (article)

This part of the Khan Academy article formalizes the two rules for preparing any quadratic equation for this method.

Read the two sections, "Rule 1" and "Rule 2". These sections provide clear worked examples for each rule, including one that results in fractions, which you should be comfortable with for the Extended paper.

The video below from Ginger Mathematician provides an excellent breakdown of how to handle these more difficult cases, which he calls "Level 3" and "Level 4". Thinking in terms of these levels can be a good way to structure your problem-solving approach.

Everything you need to know about IGCSE Completing The Square in 23 minutes (2025)

This video covers the necessary extra steps when the x^2 coefficient is not 1, or is negative.

Watch the section on "Level 3" problems starting at this timestamp. It shows how to factor out the leading coefficient before completing the square. Then, watch the brief section on "Level 4" from this timestamp, which explains the simple trick of factoring out a -1 when you have a negative x^2 term.

Conclusion

Today you've added a powerful and universally applicable tool to your algebraic toolkit. Unlike factorising, completing the square will work on any quadratic equation you encounter. It involves transforming the equation into a form where the solution can be found by simply taking a square root.

Key Takeaways:

  • Completing the square transforms an expression like x^2 + bx into a perfect square by adding the term (b/2)^2.
  • The general algorithm for solving ax^2 + bx + c = 0 is:
    1. Move the constant c to the right side.
    2. If a ≠ 1, divide the entire equation by a.
    3. Add (b/2)^2 to both sides.
    4. Factor the left side into (x + b/2)^2.
    5. Solve for x by taking the square root of both sides (remembering ±).

The process of completing the square is systematic, but it involves several steps. Wouldn't it be useful if we could just do this process once for the general quadratic equation ax^2 + bx + c = 0 and get a formula that gives us the solution directly? That is exactly what our next lesson is about. We will derive and apply the quadratic formula, which is essentially the end result of completing the square for a general quadratic.