IB Maths AA HL
Topic 2 — Functions
Paper 1 & 2
~7 min read
Factorising Quadratics
Factorising means rewriting ax2 + bx + c as a product of brackets — usually two of them. Why bother? Because once it’s factorised, the roots fall out for free: each bracket gives you one. There are three common cases you need to be fluent with — monic (leading coefficient 1), non-monic (leading coefficient bigger), and the special difference of two squares. Let me show you the practical move for each.
📘 What you need to know
- Monic quadratic x2 + bx + c: find two numbers p, q with sum = b and product = c. Then it factorises as (x + p)(x + q).
- Non-monic quadratic ax2 + bx + c: find two numbers with sum = b and product = ac. Use them to split the middle term, then factorise by grouping.
- Difference of two squares: a2x2 − c2 = (ax − c)(ax + c). Spot it when there’s no middle term and a minus sign.
- Always check for a common factor first. Pulling out a 2 or 3 makes the remaining quadratic much easier to handle.
- GDC shortcut: solve the quadratic on your calculator to get the roots, then write the factors directly. Useful on Paper 2.
Why factorise at all?
Two reasons:
Find the roots
(x − p)(x − q) = 0 ⟹ x = p or q
each bracket gives an x-intercept of the graph
Simplify expressions
cancel common brackets in fractions
essential later for rational functions and limits
Monic case — x2 + bx + c
“Monic” just means the coefficient of x2 is 1. The trick is short:
Monic factorisation
find p, q with p + q = b and pq = c
⟹ x2 + bx + c = (x + p)(x + q)
Sign rules: if c > 0, both p and q have the same sign (matching b). If c < 0, they have opposite signs.
Non-monic case — ax2 + bx + c
When a is not 1, the rule changes slightly: the product is ac instead of just c.
🧭 Recipe — non-monic by splitting the middle
- Find m, n with m + n = b and mn = ac.
- Split the middle term bx as mx + nx.
- Group the four terms in pairs and factor each pair.
- Take out the common bracket — that’s your factorisation.
If a, b, c share a common factor, pull it out first. The remaining quadratic is smaller and easier — you can often spot the factors by inspection from there.
Difference of two squares
Spot this pattern when the quadratic has no middle term and the constant is negative:
Difference of two squares
a2x2 − c2 = (ax − c)(ax + c)
Square-root each part, then write the sum-and-difference brackets. For example, 9x2 − 16 = (3x − 4)(3x + 4). No middle term needed.
The GDC shortcut
On Paper 2, this is the fastest method. Solve the quadratic on your calculator to get the roots, then write the factors:
If your GDC gives roots x = p and x = q for ax2 + bx + c = 0, then it factorises as a(x − p)(x − q). Don’t forget the a in front when a ≠ 1.
For fractional roots, clear the denominators. If x = 32 is a root, the bracket is (2x − 3). If x = −15 is a root, the bracket is (5x + 1).
Worked examples
WE 1Factorise a monic quadratic — both terms positive
Factorise x2 + 9x + 20.
Find p, q with p + q = 9 and pq = 20
try pairs of 20: 1×20, 2×10, 4×5
4 + 5 = 9 ✓ and 4 × 5 = 20 ✓
(x + 4)(x + 5)
check: x² + 5x + 4x + 20 = x² + 9x + 20 ✓
WE 2Factorise a monic quadratic with a negative constant
Factorise x2 − 5x − 14.
Find p, q with p + q = −5 and pq = −14
since pq is negative → opposite signs
try 1 & 14, 2 & 7
−7 + 2 = −5 ✓ and −7 × 2 = −14 ✓
(x − 7)(x + 2)
check: x² + 2x − 7x − 14 = x² − 5x − 14 ✓
WE 3Factorise a non-monic quadratic by splitting the middle
Factorise 6x2 − 7x − 5.
Step 1: Find m, n with m + n = −7 and mn = ac = 6 × (−5) = −30
opposite signs since mn < 0
−10 + 3 = −7 ✓ and −10 × 3 = −30 ✓
Step 2: Split the middle: −7x = −10x + 3x
6x² − 10x + 3x − 5
Step 3: Group and factor each pair
2x(3x − 5) + 1(3x − 5)
Step 4: Take out the common bracket (3x − 5)
(3x − 5)(2x + 1)
(3x − 5)(2x + 1)
check: 6x² + 3x − 10x − 5 = 6x² − 7x − 5 ✓
WE 4Factor out a common factor first
Factorise fully: 4x2 + 8x − 60.
Step 1: All terms divisible by 4 — pull it out
4(x² + 2x − 15)
Step 2: Factorise the monic quadratic inside
need p + q = 2 and pq = −15
5 + (−3) = 2 ✓ and 5 × (−3) = −15 ✓
4(x + 5)(x − 3)
always pull the common factor out first — turns a non-monic into a monic for free
WE 5Difference of two squares
Factorise: (a) 25x2 − 49, (b) 8x2 − 50.
(a) No middle term, minus sign — DOTS pattern
√(25x²) = 5x and √49 = 7
(5x − 7)(5x + 7)
(b) Pull out common factor 2 first
8x² − 50 = 2(4x² − 25)
now DOTS: √(4x²) = 2x, √25 = 5
2(2x − 5)(2x + 5)
always check for a common factor — without it, 8x² − 50 doesn’t fit DOTS cleanly
WE 6Use GDC roots to factorise
Using your GDC, the equation 10x2 + 11x − 6 = 0 has roots x = 25 and x = −32. Hence factorise 10x2 + 11x − 6.
Step 1: Convert each root into a bracket — clear denominators
x = 2/5 → 5x = 2 → bracket (5x − 2)
x = −3/2 → 2x = −3 → bracket (2x + 3)
Step 2: Multiply the brackets and check the leading coefficient matches
(5x − 2)(2x + 3) = 10x² + 15x − 4x − 6 = 10x² + 11x − 6 ✓
10x² + 11x − 6 = (5x − 2)(2x + 3)
leading coefficients of the brackets multiply to give a — that’s why clearing denominators automatically gives the right answer
💡 Top tips
- Common factor first, always. If a, b, c share a number, pulling it out simplifies everything that comes next.
- For monic quadratics, just need sum = b and product = c. List factor pairs of c and check which add to b.
- For non-monic, the product is ac not just c. Easy to forget — costs marks.
- Spot DOTS by what’s missing. No x term plus a minus sign between two perfect squares = difference of two squares.
- Always expand your answer to check. Factorising mistakes are easy to catch this way.
- Use the GDC on Paper 2. Solve the quadratic, get the roots, write the brackets — done.
- Sign rules for monic: if c > 0 the brackets have matching signs (both same as b). If c < 0 the brackets have opposite signs.
⚠ Common mistakes
- For non-monic, using product = c instead of ac. The most common slip-up.
- Forgetting the leading coefficient when reading off from GDC roots. Roots x = 2 and x = 3 don’t always mean (x − 2)(x − 3) — it could be 5(x − 2)(x − 3) if a = 5.
- Missing a common factor. 6x2 + 12x − 90 looks scary, but pulling out 6 leaves a nice monic quadratic.
- Getting signs mixed up in DOTS — it’s a difference (minus), not a sum. x2 + 9 doesn’t factor (over the reals).
- Stopping after pulling out the common factor without finishing the factorisation. 4(x2 − 9) is not fully factorised — keep going to 4(x − 3)(x + 3).
- Sign error on negative roots. If x = −3 is a root, the bracket is (x + 3), not (x − 3).
- Not checking by expansion. Catches almost all errors in seconds.
Factorising is the fastest way to find roots — but only when the quadratic factors nicely with integers. When the numbers get ugly, you need a different tool. The next note covers completing the square, which always works regardless of whether the quadratic factorises cleanly, and gives you the vertex as a bonus.
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