Two roots
x² − 5x + 6 = 0
(x − 2)(x − 3) = 0
x − 2 = 0 or x − 3 = 0
x = 2 or x = 3
Maths · Foundation and Higher · AQA · 45 minutes
Quadratic equations Why does factorising find where the curve meets the axis?
Students guess an equation for a curve, then find out why factorising finds where it meets the x-axis. Slides to present, a vote that shows who's stuck, practice and homework.
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Move c. How many times does the curve meet the x-axis?
Twice: at x = 1 and x = 3
The idea in one minute
A root is an x-value where y = 0: where the curve meets the x-axis. Written as brackets, (x − 2)(x − 3), a quadratic shows its roots, because a product is zero only when one of its factors is.
The lesson goes picture first. Students guess an equation for a curve, see why the brackets work, watch two roots become one and then none as the curve moves, and only then factorise. The core is monic quadratics, for Foundation and Higher; the stretch, 2x² + 5x − 3 = 0, is optional Higher work.
By the end, students can
- connect roots to x-intercepts
- solve a factorisable quadratic using the zero-product rule
- check both solutions by substitution
- tell two, one and no real roots apart
Key words
- quadratic
- factor
- root (solution)
- x-intercept
- repeated root
Before this lesson
Students should expand two brackets and solve a linear equation such as x − 3 = 0. Factorising monic quadratics is taught here, from worked examples.
The lesson, step by step
60 minutes in total. Each step says what changes for this length.
5 min 1 · Hook · students think
Guess my curve
Students write an equation for a curve before anyone explains it.
Say “This curve meets the x-axis at 2 and at 3. Write an equation it could have. Don't show anyone yet.”
Ask- What is y at x = 2 and at x = 3?
- Whose equation works? How could we check?
Watch for Writing y = x + 5 (from 2 + 3): ask what y is at x = 2.
For the teacher
Give two minutes of silent writing, then gather three or four equations. Test each one by putting x = 2 into it: is y zero?
Don't explain yet. The question is the point: how can an equation know where its curve meets the axis?
Gather a few guesses with Ask the class. If someone writes (x − 2)(x − 3), keep it for step 2.
Shorter: Show the curve and its equation together; skip the guessing round (−2 min).
10 min 2 · Explain · you explain
Why the brackets give the roots
A product is zero only when a factor is. Then move c: two roots, one, none.
Say “If two numbers multiply to make zero, one of them must be zero. That is the whole trick.”
Ask- Why must one bracket be zero?
- Before I move c to 4: how many times will it meet the axis?
Watch for Reading x − 2 as the root −2: set the bracket equal to zero.
One repeated root: the curve touches
x² − 4x + 4 = 0
(x − 2)² = 0
x = 2, a repeated root
No real roots: the curve stays above
x² + 2x + 5 = (x + 1)² + 4
(x + 1)² ≥ 0, so y ≥ 4
The curve never meets the x-axis.
For the teacher
Show y = (x − 2)(x − 3) and put x = 2 in: (0)(−1) = 0. Then x = 3: (1)(0) = 0. A product is zero only when one of its factors is, so the brackets give the roots. Expanded, it is x² − 5x + 6: factorising goes the other way.
On the second slide keep b = −4 and move c. At c = 3 the curve meets the axis twice; at 4 it touches once, a repeated root, (x − 2)²; at 5 it never does. Ask for a vote before each move.
For no real roots the graph is a hint, not a proof: x² + 2x + 5 = (x + 1)² + 4, and a square is never negative, so y is at least 4.
Vote before each move of c (Ask the class): at 3, then 4, then 5.
Shorter: Move c to 4 and 5 without the votes (−3 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- What is the same in the twin? What changed?
Watch for Losing x = 0 by dividing by x: take x out as a factor instead.
Two positive roots
x² − 5x + 6 = 0
(x − 2)(x − 3) = 0
x = 2 or x = 3
Then theirs
x² − 7x + 12 = 0
Roots of both signs
x² + 2x − 8 = 0
(x + 4)(x − 2) = 0
x = −4 or x = 2
Then theirs
x² + x − 6 = 0
A root at zero
x² − 5x = 0
x(x − 5) = 0
x = 0 or x = 5
Then theirs
x² − 2x = 0
For the teacher
Work the first example aloud, writing each line. Then students solve its twin while you watch who starts with the brackets.
Do the same for each pair. The third pair catches students who divide by x.
Open the answers slide once most have finished their twins.
Shorter: Do the first pair only (−3 min).
27 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 can dividing by x lose a solution?
Say “Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- How do you know no root is missing?
Watch for Giving only one root: ask whether the other bracket could be zero.
Show your factorisation and check each root by substitution. Enter the smaller root first when there are two. The stretch is optional.
The question that shows who's stuck
- Solve (x − 4)(x + 1) = 0.
In your Classroom, as they answer (an example) - A12
- B8
- C3
- 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 + 7x + 10 = 0, so x = ___ or x = ___
- x² − x − 12 = 0, so x = ___ or x = ___
- x² − 9 = 0, so x = ___ or x = ___
- x² − 6x + 9 = 0 has one solution. x = ?
- Always, sometimes or never? A quadratic curve meets the x-axis twice.
- You are given x² + 2x + 5 = (x + 1)² + 4. How many real solutions does x² + 2x + 5 = 0 have?
- Make it true: y = (x + ___)(x − ___) meets the x-axis at x = −1 and x = 4.
- The curve y = x² + bx + 12 crosses the x-axis at x = 2. So b = ___, and the other solution is x = ___
- x² − 5x = 0, so x = ___ or x = ___
- Stretch: 2x² + 5x − 3 = 0, so x = ___ or x = ___
Support · For question 1: (x + 5)(x + ___) = 0. The missing number multiplies with 5 to give 10 and adds to 5 to give 7. Then solve each bracket = 0 and check both values.
For the teacher
Everyone chooses before anyone explains. If they flip the signs, ask them to put their root back into its bracket. If they give one root, ask whether the other bracket could be zero.
Questions 1–2 build fluency; 3–4 change the look; 5–7 ask for reasons; 8 works backwards from a root; 9 keeps the root at zero. The stretch has a coefficient other than 1.
Students show their working on paper or the board; the quiz keeps their final answers.
If many chose the flipped signs, model one more bracket before they practise.
Shorter: After the vote, questions 1–4 only (−7 min).
Longer: The stretch, then: why can dividing by x lose a solution? (+10 min).
10 min 5 · Exit · students show
Sketch three, help a friend
Three sketches and one sentence show who sees roots as where y = 0.
For this length: Compare sketches: what makes each one right?
Say “Sketch three curves: two roots, one, none. Then help a friend in one sentence.”
Watch for Calling the y-intercept a root: roots have y = 0.
One sentence is enough.
- A friend says (x − 4)(x + 1) = 0 gives x = −4 and x = 1. Write them one sentence to help.
For the teacher
Sketch three upward curves: one that meets the x-axis twice, one that touches it once, one that never meets it. Label both axes.
Then answer the exit question in one sentence.
The sentences show who still flips the signs: start the next lesson with the most common mistake. Set the homework.
Longer: Compare sketches: what makes each one right? (+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 explain each choice aloud; move on when their substitution checks are secure.
Homework: solve and sketch
For questions 1–4 and 6, factorise, solve and check each root by substitution. Sketch y equal to the left-hand side and label where it meets the x-axis. For question 5, use the square given to explain why there are no real roots, then sketch.
- x² − 3x + 2 = 0
- x² + 4x + 3 = 0
- x² − 16 = 0
- x² + 6x + 9 = 0
- x² + 2x + 5 = 0. Use x² + 2x + 5 = (x + 1)² + 4.
- x² + 3x = 0
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching quadratic equations: common questions
How do you teach solving quadratics by factorising?
Picture first. Show a curve and ask for its equation, then show why the brackets work: a product is zero only when one of its factors is. Then factorise, set each factor to zero and put each root back into the equation. Start with monic quadratics before a non-monic one.
Is this quadratic equations lesson suitable for Foundation GCSE?
Yes. The core factorises monic quadratics and reads roots on graphs, for students who can expand brackets and solve linear equations. The non-monic stretch is optional Higher work, and the square in the no-roots example is explained, not assumed.
Why can a quadratic have only one real root?
When both factors are the same, they give the same root. (x − 2)² = 0 has the repeated root x = 2, and its curve touches the x-axis at (2, 0). Count different roots, not brackets.
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: A4 (factorising), A11 (roots and intercepts), A18 (solving quadratics). The non-monic stretch goes beyond the core lesson.
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