Completing the Square

OCR

GCSE

Completing the square is the fundamental algebraic process of transforming a quadratic expression $ax^2 + bx + c$ into the vertex form $a(x+p)^2 + q$. This manipulation isolates the variable within a single squared term, enabling the direct solution of quadratic equations where factorization is not possible and facilitating the derivation of the quadratic formula. Crucially, this form reveals the coordinates of the turning point (maximum or minimum) and the axis of symmetry of the parabolic graph. In advanced contexts, mastery of this technique is a prerequisite for integrating rational functions, analyzing conic sections, and manipulating complex numbers.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

What You Need to Demonstrate

Key skills and knowledge for this topic

Award M1 for a valid attempt to factorise the coefficient of $x^2$ from the first two terms only
Award M1 for correctly halving the coefficient of $x$ to form the squared bracket $(x \pm p)^2$
Award A1 for subtracting the square of $p$ and correctly multiplying by the coefficient $a$ before simplifying
Award B1 for explicitly stating the coordinates of the turning point as $(-p, q)$, ensuring correct sign inversion
Credit responses that correctly identify the nature of the turning point (maximum/minimum) based on the sign of $a$

Example Examiner Feedback

Real feedback patterns examiners use when marking

"You correctly halved the coefficient, but you forgot to multiply the subtracted term by the factor outside the bracket."
"Check your signs for the turning point coordinates; remember that $(x+3)^2$ implies $x=-3$ at the vertex."
"Good algebraic manipulation. Now, explicitly state whether this point is a maximum or minimum based on the coefficient of $x^2$."
"To secure full marks, ensure you leave the expression in the exact form requested, e.g., $a(x+p)^2 + q$."

Marking Points

Key points examiners look for in your answers

Award M1 for a valid attempt to factorise the coefficient of $x^2$ from the first two terms only
Award M1 for correctly halving the coefficient of $x$ to form the squared bracket $(x \pm p)^2$
Award A1 for subtracting the square of $p$ and correctly multiplying by the coefficient $a$ before simplifying
Award B1 for explicitly stating the coordinates of the turning point as $(-p, q)$, ensuring correct sign inversion
Credit responses that correctly identify the nature of the turning point (maximum/minimum) based on the sign of $a$

Examiner Tips

Expert advice for maximising your marks

💡When the coefficient of $x^2$ is not 1, use large square brackets to isolate the completing the square process before multiplying out by $a$
💡If asked for the 'least value' of an expression, this is simply the constant term $q$ in your final form; you do not need to use calculus
💡Always check the validity of your answer by expanding your final result—it must return exactly to the original quadratic

Common Mistakes

Pitfalls to avoid in your exam answers

Forgetting to multiply the subtracted constant term by the factor $a$ when simplifying the expression
Incorrect signs for the coordinates of the turning point, typically giving $(p, q)$ instead of $(-p, q)$
Algebraic errors when handling negative $x^2$ terms, often leading to incorrect expansion of $-1(x-p)^2$
Stopping at the algebraic form without explicitly extracting the requested coordinates or maximum/minimum value

Study Guide Available

Comprehensive revision notes & examples

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Key Terminology

Essential terms to know

Transformation to vertex form

Solving quadratic equations

Optimization (finding maxima/minima)

Geometric interpretation of parabolas

Likely Command Words

How questions on this topic are typically asked

Express

Find

Sketch

Solve

Calculate

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