National 5 Maths
Completing the Square
Page sections 
- Topic content
- Textbook page numbers
- Key ideas
- Worked examples
- Past paper questions
- Worksheets
- Notes and videos
Topic content
- Completing the square in a quadratic expression with unitary
\(x^2\) coefficient - Explanation: We need to be able to write quadratics of the form \(x^2+bx+c\) in the form \(\left(x+p\right)^2+q\), where \(b\) and \(c\) will be integers but \(p\) and \(q\) might be fractions.
Integer
The set of integers is the set of whole numbers {0,1,2,3,...} together with their negative counterparts {-1,-2,-3,...}.
Unitary
Unitary refers to the number \(1\small.\)
So a unitary quadratic (also called a monic quadratic) has \(1\) as the coefficient of \(x^2\).
Examples:
\(x^2-7x+3\)
\(x^2-5\)
\(3+8x+x^2\)
Textbook page numbers
- Zeta National 5+ Maths pp.47-51
- TeeJay Maths Book N5 pp.187-190
- Leckie National 5 Maths pp.42-46
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Key ideas
- \(\left(x+p\right)^2=x^2+2px+p^2\). The coefficient of \(x\) is double \(p\) and the constant term is the square of \(p\).
- After we complete the square, the letter \(x\) appears only once. This makes it easy to graph the parabola.
- The graph of \(y=(x+p)^2+q\) has its turning point at \((-p,q)\) and a vertical line of symmetry \(x=-p\small.\)
Constant term
A constant term in an algebraic expression is a term that doesn't depend on the variable.
Examples:
The constant term in \(i^2+i+7\) is \(7\small.\)
The constant term in \(-x^2+4\) is \(4\small.\)
The constant term in \(5-y^3\) is \(5\small.\)
The constant term in \(2+sin\ a\) is \(2\small.\)
The constant term in \(7x^2+2x\) is \(0\small.\)
Coefficient
The coefficient of a term in an algebraic expression is the constant multiple of the relevant variable.
That sounds complicated, but it isn't!
Examples:
Coefficient of \(x^2\) in \(3x^2\) is \(3\)
Coefficient of \(x^2\) in \(-x^2+4\) is \(-1\)
Coefficient of \(x\) in \(5x^2-3x\) is \(-3\)
Coefficient of \(x\) in \(6x^2-1\) is \(0\)
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Example 1 (non-calculator)
Express \(x^2+10x+28\) in the form \(\left(x+p\right)^2+q\)
Example 2 (non-calculator)
Express \(x^2+6x+1\) in the form \(\left(x+a\right)^2+b\)
Example 3 (non-calculator)
Express \(x^2-2x+5\) in the form \(\left(x+m\right)^2+n\)
Example 4 (non-calculator)
Express \(x^2-4x-9\) in the form \(\left(x+a\right)^2+b\)
Example 5 (calculator)
Express \(x^2+5x-1\) in the form \(\left(x+p\right)^2+q\)
Example 6 (non-calculator)
Express \(x^2-9x+3\) in the form \(\left(x+p\right)^2+q\)
Example 7 (non-calculator)
Find the coordinates of the turning point of the graph of \(y=x^2-8x+5\).
Example 8 (non-calculator)
SQA National 5 Maths 2014 P1 Q3
Express \(x^2-14x+44\) in the form \(\left(x-a\right)^2+b\)
Example 9 (calculator)
SQA National 5 Maths 2016 P2 Q9
Express \(x^2+8x-7\) in the form \(\left(x+a\right)^2+b\)
Example 10 (calculator)
SQA National 5 Maths 2019 P2 Q10
Express \(x^2+10x-15\) in the form \(\left(x+p\right)^2+q\)
Example 11 (non-calculator)
SQA National 5 Maths 2022 P1 Q5
(a) Express \(x^2+8x+15\) in the form \(\left(x+a\right)^2+b.\)
(b) Hence, or otherwise, state the coordinates of the turning point of the graph \(f(x)=x^2+8x+15.\)
Example 12 (calculator)
SQA National 5 Maths 2025 P2 Q5
Express \(x^2+10x+19\) in the form \(\left(x+a\right)^2+b\)
Buy N5 Maths practice papers
Zeta: Five Practice Papers TOP CHOICECGP: N5 Maths Exam Practice
Leckie: Revision and Practice
Hodder: N5 Maths Practice Papers
Past paper questions
| • All past paper questions by topic |
|
• 2014 Paper 1 Q3 • 2016 Paper 2 Q9 • 2017 Spec. P1 Q13 (with quadratics) • 2018 Paper 1 Q19 (with quadratics) • 2019 Paper 2 Q10 • 2022 Paper 1 Q5 • 2024 Paper 1 Q12 (with quadratics) • 2025 Paper 2 Q5 |
Buy our favourite N5 textbook
Zeta National 5+ Maths
Clear and comprehensive.
Progressive exercises.
Includes answers.
Buy from Zeta Press
Worksheets
|
Maths.scot worksheet • Completing the square (Answers) |
| Essential Skills worksheet • Completing the square (Answers) |
|
CJ Maths worksheet • Parabolic graphs (no answers) |
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Corbettmaths worksheets 1. Completing the square (Answers) 2. Identifying turning points (Answers) |
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National5Maths.co.uk worksheets 1. Positive coefficient (with answers) 2. Negative coefficient (with answers) |
| Larkhall Academy worksheet • Pages 30-31 Ex 5 (no answers) |
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Transum self-marking worksheets 1. Trinomials with no constant term 2. General unitary trinomials
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Buy N5 Maths revision guides
How to Pass N5 Maths TOP CHOICEBrightRED: N5 Maths Study Guide
CGP: N5 Maths Revision Guide
Notes and videos
| Videos - Maths180.com 1. Completing the square 2. Graphing quadratic functions |
| Video - Larbert Academy |
| Videos - Mr Graham Maths 1. Completing the square 2. Sketching quadratic graphs |
| Video - Mr Murray Maths Help |
| Video - Tynecastle HS Recommended: very clear teaching |
| PowerPoint - MathsRevision.com |
| Notes - MathCentre.ac.uk |
| Notes - National5.com |
| Worked examples - Maths Mutt |
| • Revision notes - BBC Bitesize • Test yourself - BBC Bitesize |
Click here to study the completing the square notes on National5.com.
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