For students and parents · O-Level E-Math & A-Math
Completing the Square, and Why the Method Beats Memorising
Rewriting a quadratic to read off its turning point.
Mrs Eileen Toh
Founder & Curriculum Architect · ex-MOE · 3 min read · Updated 29 Jun 2026
Completing the square is the algebra move that quietly answers a graph question. Rewrite a quadratic in the right form and it tells you, with no calculus at all, where the parabola turns and what its smallest or largest value is. That is why it sits on the join between two A-Math families, the Algebra Toolkit and Graphs and Functions, and why it is worth owning rather than memorising.
The idea is to turn x2 + bx + c into a perfect square plus a leftover:
x2 + bx + c = (x + b/2)2 - (b/2)2 + c.
The (x + b/2)2 part is never negative, so once the expression is in this form the minimum value is whatever is left after the square, and it occurs when the bracket is zero.
Worked example, a simple quadratic
Express x2 + 6x + 5 in completed-square form and state its minimum value.
Half of 6 is 3, so the square is (x + 3)2. That square expands to x2 + 6x + 9, which is 4 too big, so subtract 4:
x2 + 6x + 5 = (x + 3)2 - 4.
Because (x + 3)2 is never negative, the minimum value is minus 4, and it happens when x + 3 = 0, that is at x = -3. The turning point of the parabola is (-3, -4). Substituting x = -3 into the original confirms the value is minus 4.
Worked example, with a coefficient in front
Express 2x2 - 8x + 3 in completed-square form and state its minimum.
When the x2 has a coefficient, factor it out of the first two terms first:
2x2 - 8x + 3 = 2(x2 - 4x) + 3.
Complete the square inside the bracket: half of minus 4 is minus 2, so x2 - 4x = (x - 2)2 - 4. Then
2(x2 - 4x) + 3 = 2((x - 2)2 - 4) + 3 = 2(x - 2)2 - 8 + 3 = 2(x - 2)2 - 5.
The minimum value is minus 5, at x = 2. Substituting x = 2 into 2x2 - 8x + 3 confirms minus 5. Factoring the coefficient out first is the step students skip, and it is where the marks go.
Why use it at all
Completing the square reads straight off as graph features. The completed form (x + p)2 + q puts the turning point at (-p, q) and the minimum or maximum value at q, with no differentiation needed. It also sets up the quadratic formula, solves equations that will not factorise, and prepares ground for later work. One method, several questions, which is exactly the kind of high-return structure GPA prioritises.
If your child can complete the square on a plain quadratic but stumbles the moment a coefficient appears in front, that is a precise, fixable gap. GPA's Secondary Math programme teaches the method as a bridge to graphs, on real O-Level papers, and a short diagnostic consult will show you where the routine drops a step.
Both worked examples on this page were checked independently before publishing.