Complete the square
x² − 4x + 1
Half of −4 is −2: (x − 2)² = x² − 4x + 4
x² − 4x + 1 = (x − 2)² − 3
Lowest point: (2, −3)
Maths · Higher · AQA · 45 minutes
Completing the square Where is the lowest point of the curve, and how does the equation tell us?
Students find the lowest point of a curve, then see why y = (x − p)² + q has it at (p, q), and complete the square to find it and to solve. Slides to present, a vote that shows who's stuck, practice and homework.
Present and print without signing in.
Use in my lessons
Free. The whole lesson is copied into your Library, yours to change.
Before you move p, say where the lowest point will go. Why is its height always 1?
Lowest point: (2, 1)
The idea in one minute
A square is never negative, so (x − p)² + q is smallest when x = p, and then y = q: the curve's lowest point, its turning point, is (p, q). Completing the square writes x² + bx + c in that form: halve b and write (x + b ÷ 2)², then take away (b ÷ 2)² to keep it equal. The lowest point is at x = −b ÷ 2.
The lesson goes picture first. Students find the lowest point on a graph before anyone explains, then move p and q and say where it goes. Only then do they complete the square, read off the turning point, and solve equations that don't factorise. It follows the lesson on solving quadratics by factorising; the stretch has a number in front of x².
By the end, students can
- explain why y = (x − p)² + q has its lowest point at (p, q)
- complete the square for x² + bx + c
- find a quadratic's turning point by completing the square
- solve a quadratic by completing the square, exactly with surds
Key words
- completing the square
- turning point
- minimum
- surd
- ±
Before this lesson
Students should expand (x + 3)² and solve quadratics by factorising. Square roots and surds such as √3 are used; completing the square is taught here.
Taught in Quadratic equations.
The lesson, step by step
60 minutes in total. Each step says what changes for this length.
5 min 1 · Hook · students think
Find the lowest point
Students find a curve's lowest point before anyone explains.
Say “Here is y = x² − 4x + 5. Find its lowest point from the graph. Then: where in the equation could you see it?”
Ask- What is y at x = 1 and at x = 3? Why the same?
- Could the equation be written another way?
Watch for Answering with the y-intercept, (0, 5): ask whether the curve goes lower.
For the teacher
Give two minutes of silent work. Most read (2, 1) off the graph; some notice the curve is the same height either side of x = 2.
Don't explain yet. The question is the point: the equation doesn't show 2 or 1. Could another way of writing it?
Gather guesses with Ask the class. Keep (2, 1) for the next step, and anyone who tried a square.
Shorter: Show the curve and its lowest point; skip the guessing round (−2 min).
12 min 2 · Explain · you explain
Why (p, q); then completing the square
Move p and q; then write x² − 4x + 1 as (x − 2)² − 3 and solve with it.
Say “A square is never negative. So when is (x − 2)² + 1 smallest?”
Ask- Before I move p to −1: where will the lowest point be?
- Why take away 4 after writing (x − 2)²?
Watch for Adding the square of half of b instead of taking it away.
Solve with it
x² − 4x + 1 = 0
(x − 2)² = 3
x − 2 = √3 or −√3
x = 2 + √3 or x = 2 − √3
For the teacher
First slide: y = (x − p)² + q. The square is never negative and is 0 at x = p, so the lowest y is q, at x = p. Move p: the curve slides sideways. Move q: it slides up and down. The lowest point is always (p, q).
Second slide: x² − 4x + 1. Halve −4 to get −2. (x − 2)² = x² − 4x + 4, which is 3 too many, so x² − 4x + 1 = (x − 2)² − 3, lowest point (2, −3). Expand it back to check.
Third slide: to solve x² − 4x + 1 = 0, (x − 2)² = 3, so x − 2 = ±√3. The curve shows both solutions, either side of x = 2.
Vote before each move (Ask the class): p to −1, then q to −2.
Shorter: Move p only; one worked example (−4 min).
8 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
Say “Watch me do one. Then you do its twin, which is almost the same.”
Ask- How do you check your square without me?
Watch for Halving b but not squaring it.
The turning point
x² + 6x + 4
(x + 3)² = x² + 6x + 9
x² + 6x + 4 = (x + 3)² − 5
Lowest point: (−3, −5)
Then theirs
x² + 8x + 10
Real roots or not?
x² − 2x + 5 = (x − 1)² + 4
(x − 1)² + 4 is at least 4
So x² − 2x + 5 = 0 has no real roots
Then theirs
x² − 6x + 11 = 0
Solve exactly
x² + 4x − 1 = 0
(x + 2)² − 5 = 0, so (x + 2)² = 5
x = −2 + √5 or x = −2 − √5
Then theirs
x² − 6x + 4 = 0
For the teacher
Work the first example aloud, writing each line, and expand the square back out to check. Then students do its twin.
Do the same for each pair. The second pair links back to no real roots; the third catches students who take only the positive root.
Open the answers slide once most have finished their twins.
Shorter: Do the first and third pairs only (−3 min).
25 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
For this length: The stretch, then: why does a 2 in front of x² change the method?
Say “Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- Expand your answer: do you get back what you started with?
Watch for Adding the square instead of taking it away.
Show your working, and expand your square back out to check it. The stretch is optional.
The question that shows who's stuck
- x² − 6x + 2 = (x − 3)² + q. What is q?
In your Classroom, as they answer (an example) - A11
- B7
- C5
- D2
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- x² + 10x + 7 = (x + ___)² − ___
- x² − 8x + 20 = (x − ___)² + ___
- The lowest point of y = x² + 2x − 3 is (___, ___)
- Solve x² − 2x − 4 = 0 by completing the square. Give the larger solution to 2 decimal places.
- y = (x − 3)² + 5. How many times does the curve meet the x-axis?
- Which curve has its lowest point at (−2, 3)?
- x² + bx + 14 = (x + 3)² + c, so b = ___ and c = ___
- Stretch: 2x² + 12x + 7 = 2(x + ___)² − ___
Support · For question 1: half of 10 is 5, so start with (x + 5)². It expands to x² + 10x + 25. You only want + 7: what must you take away?
For the teacher
Everyone chooses before anyone explains. If they added the square, ask them to expand their answer back out. If they halved without squaring, ask what (x − 3)² expands to.
Questions 1–2 build fluency; 3–4 change the look; 5–6 ask for reasons; 7 works backwards. The stretch has 2x².
Students show their working on paper or the board; the quiz keeps their final answers.
If many added the square, or took away half of b without squaring it, expand (x − 3)² together before they practise.
Shorter: After the vote, questions 1–4 only (−6 min).
Longer: The stretch, then: why does a 2 in front of x² change the method? (+10 min).
10 min 5 · Exit · students show
Does it meet the axis?
One square completed and one reason show who links (p, q) to the roots.
For this length: Sketch it, marking the lowest point and the y-intercept
Say “Complete the square, then decide whether the curve meets the x-axis. Say how you know.”
Watch for Deciding from the sign of the number at the end of x² − 10x + 30, not from q.
Show your working; one sentence for the reason.
- Write x² − 10x + 30 in the form (x − p)² + q. Does y = x² − 10x + 30 meet the x-axis? Say how you know.
For the teacher
Write x² − 10x + 30 in the form (x − p)² + q.
Then answer the exit question: does y = x² − 10x + 30 meet the x-axis?
The reasons show who uses the lowest point to decide: start the next lesson with one that didn't. Set the homework.
Longer: Sketch it, marking the lowest point and the y-intercept (+5 min).
The answers, their slides and the mistake behind each wrong answer come with your copy. The worked examples above are open to everyone.
One-to-one: ask the student to expand their square back out every time, to check it gives the expression they started with.
Homework: turning points and solving
Show each square completed, and expand it back out to check. Give exact answers with surds for question 5.
- Complete the square for x² + 4x + 9.
- Complete the square for x² − 12x + 30.
- Find the turning point of y = x² + 6x + 5.
- Solve x² + 6x + 5 = 0 by completing the square. Check by factorising.
- Solve x² − 4x − 2 = 0, giving exact answers.
- Show that x² − 2x + 7 = 0 has no real solutions.
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching completing the square: common questions
How do you explain completing the square?
Start with the picture: y = (x − p)² + q has its lowest point at (p, q), because a square is never negative and is 0 at x = p. Completing the square rewrites x² + bx + c in that form: halve b, write the square, then take away the extra number the square brings, so the two stay equal.
How do you find a turning point by completing the square?
Write the quadratic as (x − p)² + q. The square is 0 when x = p, so the turning point is (p, q). For x² + 6x + 4 = (x + 3)² − 5, p = −3 and q = −5, so it is (−3, −5). Expand the square back out to check.
Is completing the square on the Foundation or Higher tier?
Higher. This lesson follows solving quadratics by factorising: students should be able to expand (x + 3)² and factorise first. The stretch, with 2x², goes a step further.
Can I use the lesson without creating an account?
Yes. Read the lesson, print the worksheet and open its slides without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.
Curriculum reference: AQA GCSE Mathematics 8300: A11 (turning points by completing the square) and A18 (solving quadratics by completing the square), Higher.
Teach this lesson tomorrow
Keep the whole lesson, change what you like, and use it with a group. Present its slides step by step while each student works on their own copy.
Use in my lessons
Free. The whole lesson is copied into your Library, yours to change.
In our own words; free to use with your students. Not endorsed by any exam board.