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MasterMaths online tuition: GCSE Maths Completing the Square Step by Step

GCSE Maths Completing the Square Step by Step

Writer: Ryan Harvey
Ryan Harvey
Oct 12
4 min read

Completing the square is a Higher GCSE algebra technique that rewrites a quadratic in a form revealing its turning point. Instead of memorising a rearrangement, learn why the middle coefficient determines the number inside the bracket. This step-by-step guide shows the method, the common exam traps and how to check the result by expanding.



What completing the square means


A perfect square such as (x + 3)² expands to x² + 6x + 9. The original quadratic x² + 6x + 5 is close to that expression, but has 5 instead of 9. To keep the value unchanged, write x² + 6x + 5 = (x + 3)² − 4. We have added 9 within the square and subtracted the extra 4. The two forms are exactly equal for every value of x.



GCSE Maths completing the square in four steps


For x² + 8x + 7, take half the coefficient of x: half of 8 is 4. Write (x + 4)², which expands to x² + 8x + 16. The bracket creates 16 but we need 7, so subtract 9. Therefore x² + 8x + 7 = (x + 4)² − 9. Check by expanding the square and combining the constants. A reliable check catches nearly every sign mistake before it costs marks.



How to handle a negative x coefficient


Take x² − 10x + 18. Half of −10 is −5, so begin (x − 5)². This expands to x² − 10x + 25. To return to 18, subtract 7. Therefore x² − 10x + 18 = (x − 5)² − 7. The sign inside the bracket follows the sign of half the x coefficient; the number outside corrects the constant term. Try substituting x = 0 into both expressions to make sure they agree.



Finding the minimum or maximum from the square


The form (x + 4)² − 9 is powerful because a real square cannot be negative. Its smallest value is zero, so the entire quadratic has minimum −9 when x = −4. Its turning point is (−4, −9). This is also a quick way to sketch the parabola or compare equations. When the coefficient in front of the square is positive, the graph opens upwards; a negative coefficient gives a maximum instead. Always distinguish the x-coordinate of the turning point from its y-value.



Common completing-the-square exam mistakes


The most frequent error is using the full x coefficient inside the bracket instead of halving it. Another is forgetting to square that halved number when adjusting the constant. Watch negative coefficients closely: (x − 3)² contains −6x, not −3x. Finally, some students change the original expression by adding a number without subtracting it again. A two-line expansion check is quicker than losing marks to a predictable slip.



Practice with one question at a time


Try x² + 4x + 1. Half of 4 is 2, so (x + 2)² gives x² + 4x + 4. Subtract 3, obtaining (x + 2)² − 3. Next practise x² − 6x + 11 and check that the answer is (x − 3)² + 2. Only move to coefficients other than 1 once these patterns make sense. If you repeatedly reach the wrong correction term, it may help to revisit expanding double brackets before adding exam pressure.



Frequently asked questions about GCSE Maths completing the square


Is completing the square required on Foundation GCSE?

It is normally associated with Higher-tier algebra. Check your exact exam-board specification and tier.


Why not just factorise the quadratic?

Factorising can find roots when it works conveniently; completed-square form directly shows the turning point and helps with other questions.


Explore connected maths learning topics


The following carefully selected lessons and articles support the wider skills mentioned above. Use them to find the topic that is most relevant to your current difficulty, rather than trying to read every resource at once.



Book a free online maths introduction


If your child needs a clearer explanation or you want to discuss whether one-to-one online maths tuition would be useful, MasterMaths offers an introductory conversation. Share the year group, any recent feedback and one or two questions that have caused difficulty. Ryan can explain how interactive online tuition works and help you decide on an appropriate next step. There is no promise of a fixed grade outcome; the aim is confident, independent understanding.


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