Quadratics: expand, factorise and solve
Expanding binomial products, factorising monic quadratics, and solving quadratic equations by factorising
About four lessons of 45 to 60 minutes
How do you find exactly where a ball lands, without measuring?
Throw a ball in the air and its height over time traces a curve, a parabola, described by a quadratic expression. Rearranging that expression (expanding it, or factorising it back the other way) lets you answer real questions: when does the ball hit the ground? What is the highest point? Quadratics are the algebra of anything that speeds up, slows down, or curves, not just made-up equations.
Two skills unlock this: expanding two brackets into one quadratic expression, and factorising a quadratic back into two brackets. Once you can factorise, solving a quadratic equation, finding exactly which x-values make it true, becomes almost immediate.
- The height of a kicked ball over timea parabola; solving the quadratic finds when it lands
- The area of a rectangular garden bedlength times width, expanded, gives a quadratic in one variable
- The shape of a suspension bridge cableparabolic curves appear throughout engineering and architecture
- A rocket's height during launch and descentthe two solutions of a quadratic can be liftoff and landing times
What students will be able to do
Students will expand a product of two linear binomials into a quadratic expression, factorise a monic quadratic expression into two binomials, and solve a quadratic equation with integer roots by factorising.
- I can expand (x + a)(x + b) into x^2 + (a+b)x + ab.
- I can factorise a monic quadratic x^2 + Bx + C by finding two numbers that multiply to C and add to B.
- I can solve a quadratic equation by factorising it and using the fact that if two factors multiply to zero, one of them must be zero.
- I can check a factorisation or a solution by expanding or substituting back in.
Standards this unit teaches
- AC9M9A02Australian Curriculum v9 (ACARA)Expand and factorise quadratics
Simplify algebraic expressions, expand binomial products and factorise monic quadratic expressions.
- AC9M9A04Australian Curriculum v9 (ACARA)Quadratic functions
Graph quadratic functions and solve quadratic equations graphically, numerically and, for monic quadratics with integer roots, algebraically.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Year 7 linear equations & expressions teaching unitexpanding a single bracket, a(x+b), is the stepping stone to expanding two
- Factor in the glossaryfactorising a quadratic is finding two factors that multiply back to it
- Equation in the glossarya quadratic equation is solved the same way any equation is: find the x that makes it true
Words to teach and display
- Quadratic expression
- an expression where the highest power of the variable is 2, such as x^2 + 5x + 6
- Monic
- having a coefficient of exactly 1 on the x^2 term, e.g. x^2 + ... rather than 3x^2 + ...
- Binomial
- an expression with exactly two terms, such as (x + 3)
- Factorise
- to rewrite an expression as a product of factors, the reverse of expanding
- Root (or solution)
- a value of x that makes a quadratic equation true, i.e. makes the expression equal zero
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Expanding two brackets
ConcreteTo expand (x + a)(x + b), every term in the first bracket multiplies every term in the second: x times x gives x^2, x times b and a times x both give an x term, and a times b gives the constant term.
So (x + a)(x + b) always expands to x^2 + (a + b)x + ab. The coefficient of x is the SUM of a and b, and the constant term is their PRODUCT. Spotting this pattern is the key to factorising in reverse.
Expand (x + 4)(x + 3).
- x times x = x^2.
- x times 3, plus 4 times x, gives 3x + 4x = 7x.
- 4 times 3 = 12.
Answer: (x + 4)(x + 3) = x^2 + 7x + 12.
- In (x + a)(x + b), where does the x^2 term come from?
- Why is the constant term the PRODUCT of a and b, not their sum?
2. Factorising: undoing the expansion
PictorialFactorising a monic quadratic x^2 + Bx + C reverses the expansion: find two numbers that multiply to C and add to B, and those become the two numbers inside the brackets.
For x^2 + 7x + 12, you need two numbers that multiply to 12 and add to 7. Try the factor pairs of 12: 1 and 12 (add to 13, no), 2 and 6 (add to 8, no), 3 and 4 (add to 7, yes). So it factorises as (x + 3)(x + 4), matching the expansion above in reverse.
Factorise x^2 + 8x + 15.
- Need two numbers that multiply to 15 and add to 8.
- Try factor pairs of 15: 1 and 15 (add to 16, no), 3 and 5 (add to 8, yes).
- Use 3 and 5 as the two numbers inside the brackets.
Answer: x^2 + 8x + 15 = (x + 3)(x + 5).
- What two conditions must the two numbers you pick satisfy?
- How could you check a factorisation is correct without redoing the whole search?
3. Solving quadratic equations by factorising
AbstractA quadratic equation like x^2 - x - 6 = 0 asks: which values of x make the LEFT side equal zero? Factorise first, then use a key fact: if two things multiply to zero, at least one of them must itself be zero.
x^2 - x - 6 factorises to (x - 3)(x + 2), since -3 and 2 multiply to -6 and add to -1. So (x - 3)(x + 2) = 0 means either x - 3 = 0 (giving x = 3) or x + 2 = 0 (giving x = -2). A quadratic can have up to two different solutions, and both should be checked by substitution.
Solve x^2 + 2x - 8 = 0.
- Find two numbers that multiply to -8 and add to 2: 4 and -2 work (4 x -2 = -8, 4 + -2 = 2).
- Factorise: x^2 + 2x - 8 = (x + 4)(x - 2).
- Set each factor to zero: x + 4 = 0 gives x = -4; x - 2 = 0 gives x = 2.
- Check: (-4)^2 + 2(-4) - 8 = 16 - 8 - 8 = 0. (2)^2 + 2(2) - 8 = 4 + 4 - 8 = 0. Both work.
Answer: x = -4 or x = 2.
- Why does (x + 4)(x - 2) = 0 mean x = -4 OR x = 2, not both at once?
- How many solutions can a quadratic equation have, at most?
Common misconceptions and how to address them
MisconceptionYou add the two numbers you find to get the constant term, and multiply to get the middle coefficient.
Why it happens: Students mix up which operation goes with which term when the pattern is new.
How to address it: Say it the same way every time: the two numbers ADD to the middle coefficient (B) and MULTIPLY to the constant (C). Writing B and C above the brackets while searching for the pair helps keep it straight.
Misconception(x + 3)(x + 4) = 0 means x = 3 and x = 4 at the same time.
Why it happens: Students read 'and' between the two factors instead of recognising each gives a SEPARATE, alternative solution.
How to address it: Only one factor needs to be zero for the whole product to be zero. Each bracket gives its own solution, connected by 'or', not 'and': x = -3 or x = -4 (not both together).
MisconceptionEvery quadratic factorises into two whole-number brackets.
Why it happens: Early practice only uses quadratics designed to factorise nicely, so students expect it always to work.
How to address it: Many quadratics do not factorise with whole numbers at all (their roots are irrational or not real). This unit only covers the factorisable, integer-root case; other cases need the quadratic formula, met later.
Guided practice (with answers)
1. Expand (x + 2)(x + 6).
Answer: x^2 + 8x + 12, because 2 + 6 = 8 and 2 x 6 = 12.
2. Expand (x - 3)(x + 5).
Answer: x^2 + 2x - 15, because -3 + 5 = 2 and -3 x 5 = -15.
3. Factorise x^2 + 9x + 20.
Answer: (x + 4)(x + 5), because 4 and 5 multiply to 20 and add to 9.
4. Factorise x^2 - 2x - 15.
Answer: (x - 5)(x + 3), because -5 and 3 multiply to -15 and add to -2.
5. Solve x^2 - 5x + 6 = 0.
Answer: x = 2 or x = 3, because x^2 - 5x + 6 = (x - 2)(x - 3), and setting each factor to zero gives x = 2 or x = 3.
Independent practice worksheets
Practise expanding, factorising and solving with computed, never-wrong answer keys.
Differentiation
- Start with expanding only (no negative numbers) before introducing factorising, so the pattern is secure first.
- Provide a factor-pair table (all pairs multiplying to a given number) as scaffolding while searching for the right pair.
- Colour-code B (the middle coefficient) and C (the constant) in x^2 + Bx + C so students always know which pattern they are matching.
- Use only positive-root examples at first, then introduce one negative root, then two negative roots.
- Introduce quadratics where the two numbers are not immediately obvious, requiring a longer factor-pair search.
- Ask students to construct a quadratic equation with two given integer roots, working backward from the solution to the expression.
- Explore the connection between the factorised form's roots and where the graph of the quadratic crosses the x-axis.
- Investigate what happens when a quadratic's constant term is negative versus positive, in terms of the signs of its two roots.
Assessment: exit ticket
A three-question exit ticket sampling expanding, factorising, and solving.
1. Expand (x + 5)(x + 2).
Answer: x^2 + 7x + 10, because 5 + 2 = 7 and 5 x 2 = 10.
2. Factorise x^2 + 6x + 8.
Answer: (x + 2)(x + 4), because 2 and 4 multiply to 8 and add to 6.
3. Solve x^2 - 4x - 5 = 0.
Answer: x = 5 or x = -1, because x^2 - 4x - 5 = (x - 5)(x + 1), and setting each factor to zero gives x = 5 or x = -1.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 expanding (section 1), Lesson 2 factorising (section 2), Lesson 3 solving by factorising (section 3), Lesson 4 mixed practice plus the exit ticket.
- This unit assumes comfort with expanding a single bracket, a(x + b) (Year 7 linear equations unit). Revisit that first if the distributive step is shaky.
- Language to repeat: expanding and factorising are opposite processes, and for x^2 + Bx + C the two numbers you want ADD to B and MULTIPLY to C.
- Scope note: this unit only covers monic quadratics (coefficient of x^2 is 1) with integer roots, matching AC9M9A04's explicit carve-out for 'monic quadratics with integer roots' solved algebraically. Non-monic and non-integer-root cases are out of scope here.
- Curriculum note: AC9M9A02 (Australian Curriculum v9) covers expanding and factorising; AC9M9A04 covers graphing and solving. The worked example's number-line figure previews the graphical connection (roots are x-intercepts) without requiring a full graph.
- Present and print both work: use the Print button for a clean handout, or work the factor-pair search live on the board with the class suggesting pairs.