The Completing the Square Method Explained

What Is Completing the Square?

Completing the square is a technique that transforms a quadratic expression from the form ax² + bx + c into the form a(x + p)² + q. This new form immediately tells you the coordinates of the turning point (vertex) of the parabola and makes solving certain quadratic equations much cleaner.

For the IGCSE Extended syllabus, completing the square is a key skill. It appears in questions about finding the minimum or maximum point of a quadratic function, solving equations that do not factorise neatly, and sketching parabolas.

The Basic Method (When a = 1)

Let us start with the simplest case: x² + bx + c.

Step 1: Take the coefficient of x, halve it, and square it. Step 2: Add and subtract this value inside the expression. Step 3: Group the perfect square trinomial and simplify the constants.

Example: Write x² + 6x + 2 in the form (x + p)² + q.

• Coefficient of x is 6. Half of 6 is 3. Square of 3 is 9.
• Rewrite: x² + 6x + 9 − 9 + 2
• Group: (x + 3)² − 7

So x² + 6x + 2 = (x + 3)² − 7.

The turning point of y = x² + 6x + 2 is therefore (−3, −7). Notice that p = 3 gives x = −3 (you negate p to find the x-coordinate), and q = −7 gives the y-coordinate directly.

A Shortcut Formula

For x² + bx + c, the completed square form is always:

(x + b/2)² − (b/2)² + c

This formula lets you skip the intermediate steps once you are comfortable with the method. Just halve the coefficient of x, write the bracket, then subtract the square of that half and add the original constant.

Example: x² − 10x + 18

• b/2 = −5
• (x − 5)² − 25 + 18
• (x − 5)² − 7

Turning point: (5, −7).

When a Is Not 1

When the coefficient of x² is not 1, you must factor it out first.

Example: Write 2x² + 12x + 5 in the form a(x + p)² + q.

Step 1: Factor out the coefficient of x² from the first two terms: 2(x² + 6x) + 5

Step 2: Complete the square inside the bracket: 2(x² + 6x + 9 − 9) + 5

Step 3: Distribute and simplify: 2(x + 3)² − 18 + 5 = 2(x + 3)² − 13

So the turning point of y = 2x² + 12x + 5 is (−3, −13).

The critical step that students often get wrong is distributing the factor (in this case, 2) to the constant that comes out of the bracket. When we take −9 out of the bracket that has a factor of 2 in front, it becomes −18, not −9.

Solving Equations by Completing the Square

Completing the square can be used to solve quadratic equations, especially when the quadratic does not factorise with integers.

Example: Solve x² + 4x − 3 = 0 by completing the square.

Step 1: Complete the square on the left side: (x + 2)² − 4 − 3 = 0 (x + 2)² − 7 = 0

Step 2: Rearrange: (x + 2)² = 7

Step 3: Take the square root of both sides: x + 2 = ±√7

Step 4: Solve for x: x = −2 + √7 or x = −2 − √7 x ≈ 0.646 or x ≈ −4.646 (to 3 decimal places)

Remember the ± when taking the square root — this gives you both solutions.

Connection to the Quadratic Formula

The quadratic formula itself is derived by completing the square on the general quadratic ax² + bx + c = 0. Understanding completing the square therefore gives you deeper insight into why the formula works. Some IGCSE mark schemes even give credit for deriving the formula this way.

Finding Maximum and Minimum Values

The completed square form tells you the extreme value of a quadratic function directly:

• If a > 0 (positive x² term), the parabola opens upward and the turning point is a minimum. The minimum value of the function is q, occurring at x = −p.
• If a < 0 (negative x² term), the parabola opens downward and the turning point is a maximum. The maximum value is q, occurring at x = −p.

Example: Find the maximum value of y = −x² + 8x − 10.

Factor out −1: −(x² − 8x) − 10 Complete the square: −(x² − 8x + 16 − 16) − 10 = −(x − 4)² + 16 − 10 = −(x − 4)² + 6

The maximum value is 6, occurring when x = 4.

Sketching Parabolas

The completed square form gives you three key pieces of information for sketching:

1. Turning point: (−p, q)
2. Direction: opens up if a > 0, opens down if a < 0
3. y-intercept: substitute x = 0 into the original equation

With these three pieces, plus the x-intercepts (if they exist), you can draw an accurate sketch of any parabola.

Common Mistakes

• Forgetting to halve the coefficient of x before squaring it
• Not factoring out a before completing the square when a is not 1
• Forgetting to multiply the extracted constant by a when bringing it out of the bracket
• Missing the ± when taking the square root in equation solving
• Getting the sign of the turning point x-coordinate wrong (it is −p, not p)

Practice Questions

Try these:

1. Write x² + 8x + 10 in the form (x + p)² + q. State the turning point.
2. Write 3x² − 12x + 7 in the form a(x + p)² + q.
3. Solve x² − 6x + 1 = 0 by completing the square, giving your answers to 2 decimal places.
4. Find the maximum value of y = 20 + 4x − x².

Check your answers by expanding your completed square form — it should give you back the original expression.

Get Expert Help with IGCSE Maths

Completing the square is a technique that rewards practice. Our specialist IGCSE tutors can walk you through it step by step until it becomes second nature.

Book a Free Trial Class | WhatsApp Us