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Completing the Square

HigherHigher tier onlyAQAEdexcelOCR

Master completing the square for GCSE Maths with structured, exam-style practice. This Higher resource covers completing the square and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. This rewrites x²+bx+c as (x+b/2)² − (b/2)² + c.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Completing the square rewrites a quadratic in the form \((x + p)^2 + q\). The expression is unchanged, but this form reveals information that the original hides.

Once a quadratic is in completed-square form you can read off the turning point directly: the vertex of \(y = (x + p)^2 + q\) is at \((-p,\ q)\). You can also solve the equation exactly by rearranging and taking the square root, without needing the quadratic formula.

For a quadratic \(x^2 + bx + c\), the value of \(p\) is always half of \(b\). You then subtract \(p^2\) to cancel the extra amount the squared bracket introduced, and combine it with \(c\).

Revision notes

The method for \(x^2 + bx + c\)

Halve the coefficient of \(x\) to get \(p\), then write \((x + p)^2\). Expanding this gives an unwanted \(p^2\), so subtract it and add on the original constant.

For \(x^2 + 6x + 1\): half of \(6\) is \(3\), so start with \((x + 3)^2\). That expands to \(x^2 + 6x + 9\), which is \(8\) too big, giving \((x + 3)^2 - 8\).

Reading the turning point

In the form \((x + p)^2 + q\), the turning point is at \((-p,\ q)\). The sign of \(p\) flips because the bracket is zero when \(x = -p\).

So \(y = (x + 3)^2 - 8\) has its minimum at \((-3,\ -8)\). Since a square is never negative, \(-8\) is the smallest value \(y\) can take.

Solving by completing the square

Rearrange to isolate the squared bracket, then take the square root of both sides, remembering both the positive and negative roots.

From \((x + 3)^2 - 8 = 0\) you get \((x + 3)^2 = 8\), so \(x + 3 = \pm\sqrt{8}\) and \(x = -3 \pm 2\sqrt{2}\). This gives an exact answer, which a rounded decimal would not.

Key points

  • Completed-square form is \((x + p)^2 + q\).
  • \(p\) is half the coefficient of \(x\).
  • Subtract \(p^2\), then add the original constant.
  • The turning point is at \((-p,\ q)\).
  • A minimum occurs when the \(x^2\) coefficient is positive.
  • Taking the square root gives both a positive and a negative root.

Worked examples

Example 1

Write \(x^2 + 8x + 3\) in the form \((x + p)^2 + q\).

Working

\[p = 8 \div 2 = 4\]halve the coefficient of x
\[(x + 4)^2 = x^2 + 8x + 16\]this is 16 too large, so subtract 16
\[(x + 4)^2 - 13\]subtract 16 and add the original constant 3

Example 2

Write \(x^2 - 6x + 11\) in completed-square form and state the turning point.

Working

\[p = -6 \div 2 = -3\]halve the coefficient of x, keeping the sign
\[(x - 3)^2 - 9 + 11 = (x - 3)^2 + 2\]subtract 9, then add the constant 11
\[(3,\ 2)\]the turning point is at (−p, q)

Example 3

Solve \(x^2 + 4x - 1 = 0\) by completing the square, giving an exact answer.

Working

\[(x + 2)^2 - 5 = 0\]half of 4 is 2, and −4 − 1 = −5
\[(x + 2)^2 = 5\]add 5 to both sides
\[x = -2 \pm \sqrt{5}\]take the square root of both sides, including the negative root

Common mistakes

  • Forgetting to subtract \(p^2\).

    The squared bracket introduces an extra p², so it must be removed or the expression is no longer equal to the original.

  • Getting the sign of the turning point wrong.

    For (x + 3)² − 8 the turning point is at (−3, −8), not (3, −8). The bracket is zero when x = −3.

  • Giving only the positive square root.

    A quadratic normally has two solutions, so the ± sign is essential when you take the root.

  • Rounding when an exact answer is asked for.

    Leave surds as they are. Writing 0.24 instead of −2 + √5 loses the accuracy mark.

Exam tips

  • Expand your bracket mentally to check it reproduces the original expression.
  • State the turning point as a coordinate pair when the question asks for it.
  • Keep answers in surd form whenever the question says exact.
  • If the coefficient of x² is not 1, factorise it out of the x terms first.

Key terms

Completing the square
Rewriting a quadratic in the form \((x + p)^2 + q\).
Turning point
The maximum or minimum point of a curve, where it changes direction.
Vertex
Another name for the turning point of a parabola.
Exact answer
An answer left in surd or fraction form rather than rounded.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.