Completing the Square: A Complete Guide

What Is Completing the Square?

Completing the square is a technique that rewrites a quadratic expression from the form ax² + bx + c into the form a(x + p)² + q. This new form reveals key information about the quadratic — most importantly, the coordinates of its turning point.

On the Extended IGCSE syllabus, completing the square is tested in multiple ways: as a standalone algebraic manipulation, as a method for solving quadratic equations, and as a tool for finding the minimum or maximum value of a quadratic function. Questions typically carry between three and six marks.

The Method When a = 1

When the coefficient of x² is 1, the method is most straightforward.

Step-by-Step Process

To complete the square for x² + bx + c:

1. Halve the coefficient of x: p = b/2
2. Write the perfect square: (x + b/2)²
3. Subtract the extra term: (x + b/2)² generates an unwanted (b/2)² when expanded, so subtract it
4. Add the original constant: The result is (x + b/2)² − (b/2)² + c

Worked Example 1

Complete the square for x² + 6x + 2.

• Half of 6 is 3
• Write (x + 3)²
• Expanding (x + 3)² gives x² + 6x + 9, which has an extra +9
• So x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7

Verification: Expand (x + 3)² − 7 = x² + 6x + 9 − 7 = x² + 6x + 2. Correct.

Worked Example 2

Complete the square for x² − 8x + 20.

• Half of −8 is −4
• Write (x − 4)²
• (x − 4)² = x² − 8x + 16, so there is an extra +16
• x² − 8x + 20 = (x − 4)² − 16 + 20 = (x − 4)² + 4

Worked Example 3

Complete the square for x² + 5x − 3.

• Half of 5 is 5/2
• Write (x + 5/2)²
• (x + 5/2)² = x² + 5x + 25/4, so there is an extra +25/4
• x² + 5x − 3 = (x + 5/2)² − 25/4 − 3 = (x + 5/2)² − 25/4 − 12/4 = (x + 5/2)² − 37/4

Fractions are common when the coefficient of x is odd. Do not be alarmed — the method works exactly the same way.

The Method When a ≠ 1

When the coefficient of x² is not 1, factor it out first from the x² and x terms (but not from the constant, unless you prefer a slightly different arrangement).

Step-by-Step Process

To complete the square for ax² + bx + c:

1. Factor out a from the first two terms: a(x² + (b/a)x) + c
2. Complete the square inside the bracket
3. Distribute a back through and simplify

Worked Example

Complete the square for 2x² + 12x + 5.

• Factor out 2 from the first two terms: 2(x² + 6x) + 5
• Complete the square inside: x² + 6x = (x + 3)² − 9
• Substitute: 2[(x + 3)² − 9] + 5 = 2(x + 3)² − 18 + 5 = 2(x + 3)² − 13

Negative Leading Coefficient

For expressions like −x² + 4x + 7:

• Factor out −1: −(x² − 4x) + 7
• Complete the square: x² − 4x = (x − 2)² − 4
• Substitute: −[(x − 2)² − 4] + 7 = −(x − 2)² + 4 + 7 = −(x − 2)² + 11

Be extra careful with signs when distributing the negative factor.

Finding Turning Points

The completed square form a(x + p)² + q directly reveals the turning point of the parabola:

• Turning point coordinates: (−p, q)
• If a > 0, the turning point is a minimum
• If a < 0, the turning point is a maximum

Why This Works

The expression (x + p)² is always ≥ 0. Its smallest value is 0, which occurs when x = −p.

• If a > 0: the minimum value of a(x + p)² + q is 0 + q = q, at x = −p
• If a < 0: the maximum value is q, at x = −p (because a multiplied by a positive square makes a negative contribution)

Worked Example

Find the minimum value of x² + 6x + 2 and the value of x at which it occurs.

From our earlier work: x² + 6x + 2 = (x + 3)² − 7

• Minimum value = −7
• This occurs when x + 3 = 0, so x = −3
• Turning point: (−3, −7)

Solving Equations by Completing the Square

Completing the square can also be used to solve quadratic equations:

1. Complete the square
2. Rearrange to isolate the squared bracket
3. Take the square root of both sides (remembering ±)
4. Solve for x

Worked Example

Solve x² − 4x − 1 = 0 by completing the square.

• Complete the square: (x − 2)² − 4 − 1 = 0
• (x − 2)² = 5
• x − 2 = ±√5
• x = 2 + √5 or x = 2 − √5
• x = 4.236 or x = −0.236 (to 3 d.p.)

This method is particularly useful when the question asks for answers in surd form (exact values), as it avoids the messy formula and gives clean results.

Deriving the Quadratic Formula

A beautiful application of completing the square is deriving the quadratic formula itself. Starting with ax² + bx + c = 0:

1. Divide by a: x² + (b/a)x + c/a = 0
2. Complete the square: (x + b/(2a))² − b²/(4a²) + c/a = 0
3. Rearrange: (x + b/(2a))² = b²/(4a²) − c/a = (b² − 4ac)/(4a²)
4. Square root: x + b/(2a) = ±√(b² − 4ac)/(2a)
5. Solve: x = (−b ± √(b² − 4ac))/(2a)

This derivation is sometimes asked as a “show that” question on the Extended paper.

The Line of Symmetry

The completed square form also gives the equation of the line of symmetry of the parabola:

x = −p

This is the vertical line passing through the turning point. It is useful for sketching quadratic graphs and for finding the optimal value in context problems.

Exam Tips

• Always verify by expanding. After completing the square, expand your answer to check it matches the original expression.
• Watch the signs. The most common errors come from handling negative halves and distributing negative factors. Write every step.
• Remember that (x + p)² − q has turning point (−p, −q), not (p, q). The sign of p is reversed.
• If the question says “express in the form (x + a)² + b,” complete the square. The question is telling you which technique to use.
• Practice with both integer and fractional coefficients. Exam questions frequently use odd coefficients of x, leading to fractions.

Common Mistakes to Avoid

• Forgetting to subtract (b/2)². You add (b/2)² when forming the square, so you must subtract it to maintain equality.
• Not factoring out a when a ≠ 1. The method only works directly when the x² coefficient is 1 inside the brackets.
• Distributing incorrectly. When you have a[(x + p)² − q], the result is a(x + p)² − aq, not a(x + p)² − q.
• Getting the turning point signs wrong. For (x − 3)² + 2, the turning point is (3, 2), not (−3, 2).

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