Grade 10 Factors of Quadratic Expressions: Short Notes

10 ශ්‍රේණිය · 7 පරිච්ඡේදය · වර්ගජ ප්‍රකාශනවල සාධක

85%

of Grade 10 term tests include it (46 of 54)

350

marks in past papers (67 questions)

9 min

full lesson with quick checks

🎯 What you'll learn

Factorise x² + bx + c using the product and sum of two numbers
Choose the signs of the two numbers correctly
Factorise ax² + bx + c by splitting the middle term
Take out a common factor or −1 before factorising
Check a factorisation by expanding

A carrot bed in Nuwara Eliya

A farmer in Nuwara Eliya marks out a rectangular carrot bed with area x² + 7x + 12 square metres. What are its length and breadth? Factorising the expression gives the answer: (x + 3) m and (x + 4) m. Let's learn how to find them.

Factorising undoes expanding

Expand: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12.Look closely:• 3 + 4 = 7, the coefficient of x• 3 × 4 = 12, the constant termAn expression like x² + 7x + 12 has three terms, so it is a trinomial quadratic expression. Factorising means going backwards: from x² + 7x + 12 to (x + 3)(x + 4).

The bed as four pieces

Split the bed into four parts: x², 4x, 3x and 12. Together they make the whole area, and the sides are (x + 3) and (x + 4).

The rule for x² + bx + c

Find two numbers that• multiply to give c, and• add to give b.Signs help you search:• c positive, b positive: both numbers +• c positive, b negative: both numbers −• c negative: one +, one −; the bigger one takes the sign of b

✏️ Two quick ones

Factorise (i) x² − 9x + 20 (ii) x² + 2x − 15

1(i) Product +20, sum −9: both negative. −4 × −5 = 20 and −4 + −5 = −9.
2So x² − 9x + 20 = (x − 4)(x − 5).
3(ii) Product −15, sum +2: opposite signs, bigger one +. +5 × −3 = −15 and 5 − 3 = 2.
4So x² + 2x − 15 = (x + 5)(x − 3). Expand to check.

✅ (i) (x − 4)(x − 5) (ii) (x + 5)(x − 3)

⚠️ Can't find the pair? List them all

In O/L papers many candidates could not find two numbers with product −10 and sum 3, and almost a third scored zero. Don't guess: list every factor pair of the product with signs, and add each pair.1, −10 → −9−1, 10 → 92, −5 → −3−2, 5 → 3 ✓

When x² has a coefficient

For ax² + bx + c (like 2x² + 7x + 3):1. Multiply a × c (2 × 3 = 6).2. Find two numbers with product a × c and sum b (6 and 1).3. Split the middle term with them: 2x² + 6x + x + 3.4. Factorise in pairs: 2x(x + 3) + 1(x + 3).5. Take out the common bracket: (x + 3)(2x + 1).

✏️ A term-test favourite

Factorise 3x² − 5x − 2.

1a × c = 3 × (−2) = −6. Need product −6, sum −5: that is −6 and +1.
2Split: 3x² − 6x + x − 2.
3Pairs: 3x(x − 2) + 1(x − 2).
4Common bracket: (x − 2)(3x + 1).
5Check: 3x² + x − 6x − 2 = 3x² − 5x − 2 ✓

✅ (x − 2)(3x + 1)

⚠️ Sign slip in the second pair

In 2x² + 4x − 3x − 6, the second pair is −3x − 6. Taking out −3 gives −3(x + 2), not −3(x − 2), because −3 × +2 = −6.If your two brackets don't match, check this sign first. The brackets must be identical before you can take one out.

Common factor first, always

If every term shares a factor, take it out first:2x² − 8x − 24 = 2(x² − 4x − 12) = 2(x − 6)(x + 2)If the x² term is negative, take out −1:−x² + x + 6 = −(x² − x − 6) = −(x − 3)(x + 2)Then factorise what is left inside the bracket completely.

🧮 The key idea

Find p and q, split bx into px + qx, then factorise in pairs. When a = 1 the answer is simply (x + p)(x + q).

ax² + bx + c : p + q = b , p × q = a × c

📌 Remember

• x² + bx + c: two numbers with product c and sum b.• ax² + bx + c: product a × c and sum b, split the middle term, factorise in pairs.• Take out any common factor (or −1) first.• The two brackets must match before you take one out.• Always expand your answer to check.

🧾 Quick revision sheet

Factorising x² + bx + c	Find two numbers: product c, sum b.
Factorising ax² + bx + c	Product a × c, sum b; split the middle term; factorise in pairs.
c negative: what about the signs?	One number +, one −; the bigger one has the sign of b.
c positive, b negative: signs?	Both numbers are negative.
x² − 8x + 15	(x − 3)(x − 5)
First step if all terms share a factor?	Take the common factor out first.
3x² + 2x − 5	(3x + 5)(x − 1)
How to check a factorisation?	Expand the brackets; you must get the original expression.