Study case: completing the square to find a minimum In this GCSE Completing the Square Problem Tutor, I use one study case to show how a quadratic can be rewritten so its turning point becomes clear. The problem is: write x^2 + 6x + 5 in completed square form and find its minimum value. The answer is (x + 3)^2 - 4, with minimum value -4. This is a useful GCSE and IGCSE example because it connects algebraic manipulation with graph understanding. Before solving, I ask the student what completing the square is trying to achieve. We want to rewrite the quadratic in the form (x + a)^2 + b. This form is helpful because the squared part cannot be negative. Once the expression is in this form, the minimum value can often be read directly. I also ask students to remember that completing the square is not guessing; it is based on reversing the expansion of a bracket squared. Method 1 is the standard halving method. For x^2 + 6x + 5, I take half of the coefficient of x. Half of 6 is 3, so I begin with (x + 3)^2. Expanding (x + 3)^2 gives x^2 + 6x + 9. The original expression has +5, not +9, so I subtract 4. Therefore x^2 + 6x + 5 = (x + 3)^2 - 4. I explain that the adjustment is necessary because the squared bracket created an extra 9. Method 2 is the expansion check. I ask the student to expand the answer to prove it matches the original expression. (x + 3)^2 - 4 expands to x^2 + 6x + 9 - 4, which simplifies to x^2 + 6x + 5. This check is very important because completing the square has several small steps where students can lose a sign or subtract the wrong number. A quick expansion confirms the result. Method 3 is the graph interpretation. In (x + 3)^2 - 4, the squared part is smallest when x + 3 = 0. That happens when x = -3. At that point, the expression is 0 - 4 = -4. So the minimum value is -4, and the turning point is (-3, -4). I explain that this is why completed square form is useful: it reveals the vertex of the parabola without needing a full table of values. After the three methods, I compare them with the student. The halving method gives the completed square form. The expansion check proves the algebra is correct. The graph interpretation explains the minimum value. I want students to connect all three because GCSE questions may ask for the completed square form, the turning point, the minimum value, or a sketch of the graph. The most common mistakes are forgetting to halve the coefficient of x, writing (x + 6)^2 instead of (x + 3)^2, forgetting to subtract the extra square number, and saying the minimum value is -3 instead of -4. I use these mistakes to identify what needs attention. If the student confuses the x-coordinate and y-value, I return to the idea that x = -3 makes the squared bracket zero, but the output value is -4. This topic also links to solving quadratic equations. For example, if x^2 + 6x + 5 = 0, then (x + 3)^2 - 4 = 0. This gives (x + 3)^2 = 4, so x + 3 = 2 or x + 3 = -2, giving x = -1 or x = -5. The completed square form therefore helps with graphs and equations. It is not only a separate algebra trick. A useful extension is to complete the square for x^2 - 8x + 10, where the sign in the bracket is negative. Another extension is to complete the square when the coefficient of x^2 is not 1, such as 2x^2 + 8x + 3. I introduce these only after the student is confident with the basic structure. In one-to-one tutoring, I would finish by asking the student to complete the square for a similar quadratic and explain how they know the minimum value. I want the student to say: the squared part is always zero or positive, so the lowest value occurs when the square is zero. That sentence shows real understanding. For support with this topic, students can book one-to-one tutor services: https://www.mastermathstutoring.co.uk/services. For the wider course page, visit GCSE and IGCSE Maths tutoring: https://www.mastermathstutoring.co.uk/gcse-igcse. To ask about lessons or availability, use contact MasterMaths Tutoring: https://www.mastermathstutoring.co.uk/contact-me.