IB Maths AA SL
Paper 1 & 2
14 min read
Factorising Quadratics
Factorising means rewriting a quadratic as a product of two brackets. It’s the fastest way to find the roots of a quadratic — and therefore the x-intercepts of its graph. Once you spot which case you have, the method is mechanical.
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What you need to know
- Always look for a common factor first — pull it out before anything else
- For monic quadratics (x2 + bx + c): find two numbers that add to b and multiply to c
- For non-monic quadratics (ax2 + bx + c): find two numbers that add to b and multiply to ac
- Use difference of two squares when there’s no x term and the constant is negative: a2x2 − c2 = (ax + c)(ax − c)
- On calculator papers, you can use your GDC to find the roots, then write down the factors
Why Factorise?
Factorising tells you the roots of the quadratic instantly. If you write a(x − p)(x − q) = 0, then x = p or x = q. Those values are also the x-intercepts of the graph.
The Three Cases
1
Monic
x2 + bx + c
Coefficient of x2 is 1. Find two numbers that fit.
2
Non-Monic
ax2 + bx + c
Coefficient of x2 is not 1. Same idea but use ac for the product.
3
DOTS
a2x2 − c2
Difference Of Two Squares — no x term, negative constant.
Case 1: Monic Quadratics
A monic quadratic has 1 in front of x2: e.g. x2 − 7x + 12. The strategy is to find two numbers that fit a sum and product:
Find two numbers p and q such that
SUM
p + q = b
they add to the x coefficient
PRODUCT
p × q = c
they multiply to the constant
Once you’ve found p and q, the factorised form is just (x + p)(x + q).
Spot it by inspection: if c is a small or prime number (like 5, 7, or 11), there are very few number pairs to try — you can usually see the factors instantly.
Case 2: Non-Monic Quadratics
A non-monic quadratic has a coefficient other than 1 in front of x2: e.g. 4x2 + 4x − 15. The method is similar but with one twist:
Sum
m + n = b (same as before)
Product
m × n = ac (NOT just c)
Then split the middle term bx as mx + nx and factorise by grouping:
- Check for any common factor first. If a, b, c all share a factor, pull it out.
- Find m and n with m + n = b and m × n = ac.
- Rewrite the middle term as mx + nx.
- Factorise in two pairs (group the first two terms, group the last two).
- Pull out the common bracket — that’s your factorisation.
Case 3: Difference of Two Squares (DOTS)
A special shortcut. Use it when:
- There’s no x term (the middle term is missing)
- The constant is negative
The formula:
a2x2 − c2 = (ax + c)(ax − c)
Just square root both terms — and write down the sum and difference of those roots.
DOTS in action
9x2 − 16
→
√9 = 3, √16 = 4
→
(3x + 4)(3x − 4)
Important: DOTS only works when there’s a minus sign between the two squares. a2x2 + c2 does not factorise (with real numbers).
Worked Examples
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Example 1 — Monic, simple
Factorise x2 + 7x + 10 fully.
Answer:
Find p and q with p + q = 7 and p × q = 10.
2 + 5 = 7 ✓
2 × 5 = 10 ✓
Write down (x + p)(x + q).
(x + 2)(x + 5)
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Example 2 — Monic with negatives
Factorise x2 − 7x + 12 fully.
Answer:
Find p and q with p + q = −7 and p × q = 12.
Both numbers must be negative (sum negative, product positive).
−3 + (−4) = −7 ✓
(−3) × (−4) = 12 ✓
Write down (x + p)(x + q).
(x − 3)(x − 4)
Quick check: expand (x−3)(x−4) = x² − 7x + 12 ✓
Factorise 4x2 + 4x − 15 fully.
Answer:
Step 1: find m and n with m + n = b = 4 and m × n = ac = 4 × (−15) = −60.
10 + (−6) = 4 ✓
10 × (−6) = −60 ✓
Step 2: split the middle term as 10x − 6x.
4x² + 10x − 6x − 15
Step 3: factorise in pairs.
2x(2x + 5) − 3(2x + 5)
Step 4: pull out the common bracket (2x + 5).
(2x − 3)(2x + 5)
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Example 4 — Difference of two squares (with common factor)
Factorise 18 − 50x2 fully.
Answer:
Step 1: pull out the common factor of 2.
2(9 − 25x²)
Step 2: spot DOTS — no x term, negative constant.
9 = 3², 25x² = (5x)²
Step 3: apply DOTS — sum and difference of square roots.
2(3 − 5x)(3 + 5x)
Always factor the common term first — otherwise you’d miss the simplification.
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Example 5 — Using a GDC to factorise
Use your GDC to help factorise 6x2 + x − 2.
Answer:
Step 1: solve 6x² + x − 2 = 0 on your GDC.
x = −2/3 or x = 1/2
Step 2: turn each root into a factor by clearing the denominator.
x = −2/3 → 3x = −2 → 3x + 2 = 0 → factor (3x + 2)
x = 1/2 → 2x = 1 → 2x − 1 = 0 → factor (2x − 1)
(3x + 2)(2x − 1)
Check: leading coefficient 3 × 2 = 6 ✓, constant 2 × (−1) = −2 ✓
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Tips
- Always look for a common factor first. 3x2 + 6x − 45 → factor 3 first to get 3(x2 + 2x − 15) — much easier to factorise.
- Sign of the product tells you a lot. If c is positive, both numbers have the same sign. If c is negative, the numbers have opposite signs.
- Sign of the sum tells you the rest. Once you know whether the signs match, the sign of b tells you which side dominates.
- Use the GDC shortcut on Paper 2 — solve = 0, get the roots, then write the factors. Saves time and avoids algebra mistakes.
- Always check by expanding. Expand your factors and verify they give back the original quadratic. Takes 5 seconds.
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Common mistakes
- Forgetting to factor out the common factor first. 2x2 + 8x + 6 — pull out 2 first, then factorise. Otherwise you make life harder.
- For non-monic, using c instead of ac for the product. The two numbers must multiply to ac, not c.
- Sign errors with the two numbers. x2 − 5x + 6 needs (x − 2)(x − 3), not (x + 2)(x + 3).
- Trying to use DOTS on a sum. x2 + 9 does not factorise — you need a minus sign between the squares.
- Stopping at “almost factorised.” If you get something like (2x − 4)(x + 3), pull the 2 out: 2(x − 2)(x + 3). “Fully factorise” means fully.
- Confusing roots with factors. If a root is x = 3, the factor is (x − 3), not (x + 3). The sign flips.
Final word: Three cases — monic, non-monic, DOTS — and one shortcut: the GDC. Match the case, find your two numbers, write down the brackets. Practice gives speed.
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